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AUTHOR: 


KEYNES,  JOHN  NEVILLE, 

1852-1949 


TITLE: 


STUDIES  AND 
EXERCISES  IN 

PLACE: 

LONDON 

DA  TE : 

1884 


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Original  Material  as  Filmed  -  Existing  Bibliographic  Record 


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Keynes,  John  Neville,  1852-1^^^« 

Studies  and  exercises  in  formal  logic,  includ- 
ing a  genernlisation  of  logical  processes  in 
their  application  to  complex  inferences... 
London,  Macmillan,  1884. 

ix,  414  p.   tables,  diagrs.   19^  cm. 


Restrictions  on  Use: 


160 
K52 


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2d  ed...  enl.   London,  Macmillan,  1887. 


'^  xii,  466  p.   23  cm. 

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xviii^.476  p. 


London,  1894. 


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STUDIES   AND    EXERCISES 


IN 


FORMAL    LOGIC. 


STUDIES   AND    EXERCISES 


IN 


FORMAL    LOGIC 


INCLUDING 


A   GENERALISATION    OF    LOGICAL    PROCESSES    IN    THEIR 
APPLICATION   TO   COMPLEX   INFERENCES, 


BY 


JOHN    NEVILLE   KEYNES,    M.A., 

LATE   FELLOW   OK    PEMBROKE   COLLEGE,    CAMBRIDGE. 


Honlron : 

MACMILLAN    AND    CO. 

1884 

[  The  Right  of  Translation  and  Reproduction  is  reserved.^ 


n 


PREFACE. 


Cambritige : 

PRINTED     BY    C.   J.    CLAY,    M.A.    &.   SON, 
AT  THE   UNIVERSITY   PRESS. 


FROM  THE  LIBRARY  OF 


In  addition  to  problems  worked  out  in  detail  and 
unsolved   problems,  by  means  of  which  the  student 
may  test  his  command  over  logical  processes,  the  fol- 
lowing pages  contain  a  somewhat  detailed  exposition 
of  certain  portions  of  what  may  be  called  the  book- 
work  of  Formal  Logic.     This  was  necessary  in  the 
case  of  disputed  or  doubtful  points  in  order  that  the 
working  out  of  the  problems  might  be  made  consistent 
and  intelligible;  there  were  also  some  points  concern- 
ing which  I  was  dissatisfied  with  the  method  of  treat- 
ment  adopted   in  the  ordinary  text-books.     At  the 
same  time,  this  volume   must   be   regarded,  not   as 
superseding  the  study  of  an  elementary  text-book  of 
Formal  Logic,  but  as  supplementing  it.     While  cer- 
tain topics  are  dealt  with  in  considerable  detail  others 
have  been  omitted  ;  e.^.,  the  doctrines  of  Definition 
and   Division   and   the   Predicables    are  not  touched 
upon,  no  definition  of  the  Science  itself  is  given,  and 
no  systematic  discussion  of  first  principles  has  been 

K.  L.  / 


VI 


PREFACE. 


introduced.  For  a  general  outline  of  my  views  on 
the  position  of  Formal  Logic  I  may  refer  the  reader 
to  an  article  in  Mind  for  July,  1879.  For  several 
reasons  I  should  have  been  glad  to  rewrite  and  in 
some  respects  to  modify  this  paper;  but  anything 
like  an  adequate  treatment  of  the  subject  would  have 
enlarged  the  book  considerably  beyond  the  limits  that 
I  had  assigned  to  it. 

I  have  not  endeavoured  to  distinguish  definitely 
between  book-work  and  problem ;  and  the  unanswered 
exercises  are  not  separated  and  placed  apart  at  the 
end  of  the  chapters,  but  arc  introduced  at  the  points 
at  which  the  student  who  is  systematically  working 
through  the  book  will  find  himself  in  a  position  to 
solve  them.  Exercises  of  a  similar  character  have 
not  been  to  any  considerable  extent  multiplied,  but  I 
believe  that  no  kind  of  problem  relating  to  the  opera- 
tions of  Formal  Logic  has  been  overlooked.  By 
reference  to  sections  261,  262,  281—285,  the  reader 
will  find  that  the  ordinary  syllogism  admits  of  pro- 
blems of  some  complexity. 

In  the  expository  portions  of  Parts  T.  II.  and  III., 
dealing  respectively  with  Terms,  Propositions,  and 
Syllogisms,  I  have  in  the  main  followed  the  tradi- 
tional lines,  though  with  a  few  modifications  ;  e.g.,  in 
the  systematization  of  immediate  inferences,  and  in 
some  points  of  detail  in  connection  with  the  syllogism 
to  which  I  need  not  make  further  reference  here. 
For  purposes  of  illustration  Euler's  diagrams  are  em- 


PREFACE. 


Vll 


\ 


ployed  to  a  greater  extent  than  is  usual  in  English 
manuals. 

In   Part  IV.,  which  contains   a   generalisation    of 
logical  processes  in  their  application  to  complex  in- 
ferences, a  somewhat  new  departure  is  taken.     So  far 
as  I  am  aware  this  part  constitutes  the  first  systematic 
attempt  that  has  been  made  to  deal  with  formal  rea- 
sonings of  the  most  complicated  character   without 
the  aid  of  mathematical  symbols  and  without  aban- 
doning  the   ordinary   non-equational    or   predicative 
form  of  proposition.     In    this  attempt    I   have   met 
with  greater  success  than  I  had  anticipated;  and  I 
believe  that  the  methods   which   I   have  formulated 
will  be  found   to  be   as  easy  of  application   and  as 
certain    in    obtaining   results    as    the    mathematical, 
symbolical,    or    diagrammatic    methods    of    Boole, 
Jevons,  Venn    and    others.     The  reader  may  judge 
of  this  for  himself  by  comparing  with   Boole's  ovvn 
solutions    the    problems    discussed    in    sections    368, 
369,  383— 3S6;  or  by  solving  by  different  methods 
other   of  the  problems,  e.g.,  the  very  complex  one 
contained  in  section  408.     The  book  concludes  with 
a  general  method  of  solution  of  what  Professor  Jevons 
called  the  Inverse  Problem,  and   which   he   himself 
seemed    to    regard    as    soluble  only    by    a    series  of 
guesses.  . 

Of  the  Questions  and  Problems  more  than  half 
are  my  own  composition.  Of  the  remainder,  about 
a    hundred    have    been    taken    from   various    exami- 


VIU 


PREFACE. 


nation  papers,  and  about  sixty  are  from  the  published 
writings  of  Boole,  De  Morgan,  Jevons,  Solly,  Venn  and 
Whately.  In  the  latter  case  the  name  of  the  author 
is  appended,  generally  with  a  reference  to  the  work 
from  which  the  example  is  taken.  In  the  case  of 
problems  selected  from  examination  papers,  a  letter 
is  added  indicating  their  source,  as  follows: — C.= 
University  of  Cambridge  ;  L.  =  University  of  London  ; 
N.  =  J.  S.  Nicholson,  Professor  of  Political  Economy 
in  the  University  of  Edinburgh  ;  O.  =  University  of 
Oxford  ;  R.  =  G.  Croom  Robertson,  Professor  of  Mental 
Philosophy  and  Logic  in  University  College,  London; 
V.  =  J.  Venn,  Fellow  and  Lecturer  of  Gonvillc  and 
Caius  College,  Cambridge;  W.=  J.  Ward,  Fellow  and 
Assistant  Tutor  of  Trinity  College,  Cambridge. 

The  logicians  to  whom  I  have  been  chiefly  in- 
debted are  De  Morgan,  Jevons  and  Venn.  De  Mor- 
gan's various  logical  writings  are  rendered  somewhat 
formidable  and  uninviting  by  reason  of  the  multipli- 
cation of  symbols  and  formulae  which  he  is  never 
tired  of  introducing,  and  this  is  probably  the  reason 
why  they  are  little  read  at  the  present  time  ;  they 
nevertheless  constitute  a  mine  of  wealth  for  all  who 
are  interested  in  the  developments  of  Formal  Logic. 
With  Jevons  I  have  continually  found  myself  in  dis- 
agreement on  points  of  detail,  and  it  is  possible  that  I 
may  give  the  impression  of  having  taken  up  a  special 
position  of  antagonism  with  regard  to  him.  This  is 
far  from  being  really  the  case.     I  believe  that  since 


PREFACE. 


IX 


Mill  no  one  else  has  given  such  an  impetus  to  the 
study  of  Logic,  and  I  hold  that  in  more  than  one 
direction  he  has  led  the  way  in  new  developments  of 
the  science  that  are  of  great  importance. 

To  Mr  Venn  I  am  peculiarly  indebted,  not  merely 
by  reason  of  his  published  writings,  especially  his  Sym- 
bolic Logic,  but  also  for  most  valuable  suggestions 
and  criticisms  given  to  me  while  this  book  was  in 
progress.  I  am  glad  to  have  this  opportunity  of  ex- 
pressing to  him  my  thanks  for  the  ungrudging  help 
he  has  afforded  me.  I  am  also  under  great  obliga- 
tion to  Miss  Martin  of  Ncwnham  College  and  to 
Mr  Caldecott  of  St  John's  College  for  criticisms  which 
I  have  found  very  helpful. 


6,  Harvey  Road,  Cambridge, 
19  January,  1884. 


CONTENTS. 


PART   I. 
TERMS. 

CMAPTFR  PAGE 

I.  General   and   Singular  Names.     Concrete  and  Abstract 

Names i 

II.  Connotation  and  Denotation     .         .         .         .         .         '13 

III.  Positive  and  Negative  Names.     Relative  Names      .         .       27 


PART   II. 


PROPOSITIONS. 


I.  Kinds   of  Propositions.     The   Quantity  and  Quality  of 

Propositions      ....... 

II.  The  Opposition  of  Propositions        .... 

III.  The  Conversion  of  Propositions       .... 

IV.  The  Obversion  and  Contraposition  of  Propositions 
V.  'i  ne  Inversion  of  Propositions  .... 

\T.  The  Diagrammatic  Representation  of  Propositions 

VII.  The  Logical  Foundation  of  Immediate  Inferences    . 

VIII.  Predication  and  "  Existence"  .... 

IX.  Hypothetical  and  Disjunctive  Propositions 


35 
55 
70 
76 
%6 
96 
no 
116 

130 


XIl 


CONTENTS. 


PART  III. 


SYLLOGISMS. 


CHAPTER 

I.  The  Rules  of  the  Syllogism      .... 

II.  Simple  Exercises  on  the  Syllogism    . 

III.  The  Figures  and  Moods  of  the  Syllogism 

IV.  The  Reduction  of  Syllogisms    .... 
V.  The  Diagrammatic  Representation  of  Syllogisms 

VI.  Irregular  and  Compound  Syllogisms 

VII.  Hypothetical  Syllogisms           .... 
VIII.  Disjunctive  Syllogisms 

IX.  The  Quantification  of  the  Predicate 

X.  Examples  of  Arguments  and  Fallacies      . 

XL  Problems  on  the  Syllogism        .... 


PACK 
140 
161 
167 

202 
211 
227 

246 
259 
273 


PART  IV. 

A  GENERALISATION  OE  LOGICAL  PROCESSES 
IN  THEIR  APPLICATION  TO  COMPLEX  PRO- 
POSITIONS, 


I.  The  Combination  of  Simple  Terms 

II.  The  Simplification  of  Complex  Propositions     . 

III.  The  Conversion  of  Complex  Propositions 

IV.  The  Obversion  of  Complex  Propositions 

V.  The  Contraposition  of  Complex  Propositions  . 

VI.  The  Combination  of  Complex  Propositions      . 

VII.  Inferences  from  Combinations  of  Complex  Propositions 
VIII.  Problems  involving  three  terms 

IX.  Problems  involving  four  terms 

X.  Problems  involving  five  terms 

XI.  Problems  involving  six  or  more  terms 
XII.  Inverse  Problems     .... 


290 
296 

3" 
314 
318 
335 
3.S9 
34^> 
352 

3^>3 
38+ 
395 


STUDIES  AND  EXERCISES  IN 
FORMAL  LOGIC. 


PART   I. 

TERMS. 


CHAPTER   I. 

GENERAL   AND    SINGULAR   NAMES.       CONCRETE   AND 

ABSTRACT    NAMES. 

1.    Brief  definitions  of  word,  name,  term,  symbol, 

concept. 

A  word  is  an  articulate  sound,  or  the  written  equivalent 
of  an  articulate  sound,  which  either  by  itself  or  in  con- 
junction with  other  words,  constitutes  a  name,  or  forms  a 

sentence. 

A  navie  is  "  a  word  taken  at  pleasure  to  serve  for  a 
mark  which  may  raise  in  our  mind  a  thought  like  to  some 
thought  we  had  before,  and  which  being  pronounced  to 
others,  may  be  to  them  a  sign  of  what  thought  the  speaker 
had  or  had  not  before  in  his  mind."     Hohbes. 

A  term  is  a  name  regarded  as  the  subject  or  the  predi- 
cate of  a  proposition. 

K.  L.  ' 


2  TERMS.  [part  I. 

A  symbol,  in  its  widest  signification,  is  a  sign  of  any- 
kind  ;  narrowing  our  point  of  view,  it  is  any  written  sign ; 
and  narrowing  it  still  more,  it  is  a  written  sign  which  is 
employed  without  the  realization  at  each  step  of  its  full 
signification.  Thus,  when  symbols  are  used  in  algebraical 
reasoning,  it  is  according  to  certain  fixed  rules,  without 
reference  to  or  thought  of  their  ulterior  meaning.  Names 
may  themselves  be  employed  as  symbols  in  this  sense.  Of 
course,  in  the  widest  sense,  all  names  are  symbols. 

A  concept  is  defined  by  Sir  William  Hamilton  as  "  the 
cognition  or  idea  of  the  general  character  or  characters, 
point  or  i)oints,  in  which  a  plurality  of  objects  coincide." 
In  other  words,  a  concept  is  the  mental  equivalent  of  a 
general  name. 

2.    Catcgorcmatic  and  Syncategorematic  Words. 

A  categorcmatic  word  is  one  which  can  by  itself  be  used 
as  a  term,  /.  ^.,  which  can  stand  alone  as  the  subject  or  the 
predicate  of  a  proposition. 

A  syncategorcmatic  word  is  one  which  cannot  by  itself  be 
used  as  a  term,  but  only  in  combination  with  one  or  more 
other  words. 

Any  noun  substantive  in  the  nominative  case,  or  any 
other  part  of  speech  employed  as  equivalent  to  a  noun 
substantive,  may  be  used  categorematically. 

Adjectives  are  sometimes  said  to  be  used  categore- 
matically by  a  grammatical  ellipsis.  In  the  examples, 
**The  rich  are  happy,"  *'Blue  is  an  agreeable  colour," 
either  a  substantive  is  understood  as  being  qualified  by  the 
adjective,  or  the  adjective  is  used  as  a  substantive,  that  is, 
as  a  mark  of  something,  not  merely  as  a  mark  qualifying 
something. 

Any  part  of  speech,  or  the  inflected  cases  of  nouns 


CHAP,  I.] 


TERMS. 


substantive,  may  be  used  categorematically  by  a  suppositio 
materialise  that  is,  by  speaking  of  the  mere  word  itself  as  a 
thing;  for  example,  "John's  is  a  possessive  case,*'  "Rich  is 
an  adjective,"  *'With  is  an  English  word." 

Using  the  word  term  in  the  sense  in  which  it  was  defined 
in  the  preceding  section,  it  is  clear  that  we  ought  not  to 
speak  of  syncategorematic  terms, 

3.     General,  singular  and  proper  names. 

A  general  name  is  a  name  which  is  capable  of  being 
truly  affirmed,  in  the  same  sense,  of  each  of  an  indefinite 
number  of  things,  real  or  imaginary.  A  singular  name  is  a 
name  which  is  capable  of  being  truly  affirmed,  in  the  same 
sense,  of  only  one  thing,  real  or  imaginary.  K  proper  name 
is  a  singular  name  given  merely  to  distinguish  an  individual 
person  or  thing  from  others,  its  application  after  it  has  been 
once  given  being  independent  of  any  special  attributes  that 
the  individual  may  possess  \ 

Thus,  Prime  Minister  o/Etigland  is  a  general  name,  since 
at  different  times  it  may  be  applied  to  different  individuals. 
We  may,  for  example,  talk  about  "  the  prime  ministers  of 
England  of  the  present  century."  The  name  is  however 
made  singular  by  the  prefix  "///^,"  meaning  "the  present 
prime  minister,"  or  "  the  prime  minister  at  the  time  to  which 
we  are  referring."  Similarly  any  general  name  may  be  made 
singular ;  for  example,  man,  the  first  man  ;  star,  the  pole  star. 

The  name  God  is  singular  to  a  monotheist  as  the 
name  of  the  Deity,  general  to  a  polytheist,  or  as  the 
name   of  anything  worshipped   by  anybody.      Universe  is 

^  A  proper  name  might  perhaps  be  defined  as  "a  non-connotative 
singular  name."  But  this  definition  presupposes  a  distinction  which  is 
best  given  subsequently,  and  it  would  give  rise  to  a  controversy,  that 
also  had  better  be  postponed.    Compare  section  14. 

I 2 


TERMS. 


[part  1. 


general  in  so  far  as  we  distinguish  different  kinds  of 
universes,  e.g.,  the  material  universe,  the  terrestrial  universe, 
&c. ;  it  is  singular  if  we  mean  the  universe.  Space  is 
general  if  we  mean  a  particular  portion  of  space,  singular 
if  we  mean  space  in  the  aggregate.  Water  is  general. 
Professor  Bain  takes  a  different  view  here;  he  says,  "Names 
of  Material — earth,  stone,  salt,  mercury,  water,  flame, — are 
singular.  They  each  denote  the  entire  collection  of  one 
species  of  material"  {Logic,  Deduction,  pp.  48,  49).  But 
when  we  predicate  anything  of  these  terms  it  is  generally  of 
any  portion  (or  of  some  particular  portion)  of  the  material 
in  question,  and  not  of  the  entire  collection  of  it  cojisidered 
as  one  aggregate;  thus,  if  we  say,  "Water  is  composed  of 
oxygen  and  hydrogen,"  we  mean  any  and  every  particle  of 
water,  and  the  name  has  all  the  distinctive  characters  of  the 
general  name.  Similarly  with  regard  to  the  other  terms 
mentioned  in  the  above  quotation.  It  is  also  to  be  ob- 
served that  we  distinguish  different  kinds  of  stone,  salt,  &c. 
A  name  is  to  be  regarded  as  general  if  it  may  be  poten- 
tially affirmed  of  more  than  one,  although  it  accidentally 
happens  that  as  a  matter  of  fact  it  can  be  actually  affirmed 
of  only  one,  e.g.,  King  of  England  and  Spain.  We  must 
also  note  the  case  in  which  we  are  dealing  with  a  name 
that  actually  is  not  applicable  to  any  individual  at  all ;  e.g.^ 
President  of  the  British  Rcpiddic.  A  really  singular  name  is 
distinguished  from  these  by  not  being  even  potentially 
applicable  to  more  than  one  individual;  e.g.,  the  last  of  the 
Mohicans,  the  eldest  son  of  King  ILdward  the  First^. 

^  It  seems  desirable  to  make  the  distinction  implied  in  this  para- 
graph; still  I  am  not  sure  that  it  might  not  in  some  cases  be  very 
dilTicult  to  apply  it  satisfactorily.  Nearly  all  these  divisions  of  names 
tend  to  give  rise  in  the  last  resort  to  metaphysical  difificulties;  but,  in 
my  opinion,  these  should  as  far  as  possible  be  avoided  in  a  logical 
treatise. 


CHAP.  I.] 


TERMS. 


Victoria  is  the  name  of  more  than  one  individual,  and 
can  therefore  be  truly  affirmed  of  more  than  one  individual. 
Is  it  therefore  general?  Mill  answers  this  question  in  the 
negative,  and  rightly,  on  the  ground  that  the  name  is  not 
here  aftirmed  of  the  different  individuals  in  the  same  sense. 
Professor  Bain  brings  out  this  distinction  very  clearly  in  his 
definition  of  a  general  name:  "A  general  name  is  applicable 
to  a  number  of  things  in  virtue  of  their  being  similar,  or 
having  something  in  common."  Victoria  is  then  not  general 
but  singular;  and  it  belongs  to  the  sub-class  of  proper 
names. 

4.  Collective  Names ;  and  the  collective  use  of 
names.     Are  all  collective  names  singular  } 

A  collective  name  is  one  which  is  the  name  of  a  group 
of  things  considered  as  one  whole ;  e.g.,  regiment,  nation, 
army. 

A  collective  name  may  be  singular  or  general.  It  is 
the  name  of  a  group  or  collection  of  things,  and  so  far 
as  it  is  capable  of  being  truly  affirmed  in  the  same  sense 
of  only  one  such  group,  it  is  singular;  e.g.,  the  29th  regi- 
ment of  foot,  the  English  nation,  the  Bodleian  library.  But 
if  it  is  capable  of  being  truly  affirmed  in  the  same  sense  of 
each  of  several  such  groups  it  is  to  be  regarded  as  general; 
e.g.,  regiment,  nation,  library.  Professor  Bain  writes  as  if  a 
name  could  be  general  and  singular  at  the  same  time, — 
"Collective  names  as  nation,  army,  multitude,  assembly, 
universe,  are  singular;  they  are  plurality  combined  into 
unity.  But,  inasmuch  as  there  are  many  nations,  armies, 
assemblies,  the  names  are  also  general.  There  being  but 
one  'universe',  that  term  is  collective  and  singular".  I  should 
rather  say  that  as  the  above  stand,  with  the  possible  excep- 
tion of  universe,  they  are  not  singular  at   all.      Mill   and 


TERMS. 


[part  I. 


others  imply  that  there  is  a  distinction  between  collective 
and  general  names.  The  real  distinction  however  is  be- 
tween the  collective  and  distributive  use  of  names.  A  col- 
lective name  such  as  nation,  or  any  name  in  the  plural 
number,  is  the  name  of  a  collection  or  group  of  things. 
These  we  may  regard  as  one  whole,  and  something  may 
be  predicated  of  them  that  is  true  of  them  only  as  a 
whole;  in  this  case  the  name  is  used  collectively.  On  the 
other  hand,  the  group  may  be  regarded  as  a  series  of  units, 
and  something  may  be  predicated  of  these  which  is  true 
of  them  only  taken  individually;  in  this  case  the  name 
is  used  distribiitively.  Also,  when  anything  is  predicated 
of  a  series  of  such  groups  tlie  name  is  used  distributively. 

The  above  distinction  may  be  illustrated  by  the  pro- 
positions,— All  the  angles  of  a  triangle  are  equal  to  two 
right  angles,  All  the  angles  of  a  triangle  are  less  than  two 
right  angles.  The  subject  term  is  the  same  in  both  these 
cases,  but  in  the  first  case  the  predication  is  true  only  of 
the  angles  all  taken  together,  while  in  the  second  it  is  true 
only  of  each  of  them  taken  separately ;  in  the  first  case 
therefore  the  term  is  used  collectively,  in  the  second  dis- 
tributively. 

The  peculiarity,  then,  of  a  collective  name  is  that  it  can 
be  used  collectively  in  the  singular  number,  while  other 
names  can  be  used  collectively  only  in  the  plural  number ; 
compare,  for  example,  the  names  'clergyman'  and  'the 
Clergy.'  Collective  names  in  the  plural  number  may  them- 
selves be  used  distributively,  and  it  is  therefore  not  correct 
to  say  that  all  collective  names  are  singular.  It  may  indeed 
be  held  that,  while  this  is  true,  still  when  a  name  is  used 
collectively,  it  is  equivalent  to  a  singular  name.  For 
example.  The  whole  army  was  annihilated,  The  mob  filled 
the  square.     But  I  am  doubtful  whether  even  this  is  true  in 


CHAP.  I.] 


TERMS. 


such  a  case  as  the  following,— In  all  cases  all  the  angles 
of  a  triangle  are  equal  to  two  right  angles. 

5.  Select  the  terms  that  are  used  collectively  in 
the  following  propositions;  also  classify  the  terms 
contained  in  these  propositions  according  as  they 
are  collective,  singular,  and  general  respectively,  and 
find  in  what  way  these  classes  overlap  one  another: — 

The  Conservatives  are  in  the  majority  in  the 
House  of  Lords. 

All  the  tribes  combined. 

The  nations  of  the  earth  rejoiced. 

Crowds  filled  all  the  churches. 

One  generation  passeth  away  and  another  genera- 
tion Cometh. 

Your  boxes  weigh  140  lbs. 

The  volunteers  mustered  in  considerable  numbers. 

Time  flies. 

True  poets  arc  rare. 

Those  who  succeeded  were  few  in  number. 

The  mob  was  soon  dispersed. 

Our  armies  swore  terribly  in  Flanders. 

The  multitude  is  always  in  tne  wrong. 

6.    Abstract  and  Concrete  Names. 

Mill  defines  abstract  and  concrete  names  as  follows  : 

"A  concrete  name  is  a  name  which  stands  for  a  thing; 
an  abstract  name  is  a  name  which  stands  for  an  attribute 
of  a  thing"  (Z^.^/r,  i.  p.  29)'.  In  many  cases,  this  distinction 
is  of  easy  application;  for  example,  triangle  is  the  name  of 
something  that  possesses  the  attribute  of  being  bounded  by 

1  The  references  are  to  the  ninth  edition  of  Mill's  Logic. 


8 


TERMS. 


[part  I. 


three  straight  lines,  and  it  is  a  concrete  name;  triangularity 
is  the  name  of  this  distinctive  attribute  of  triangles,  and  it 
is  an  abstract  name.  But  there  are  other  cases  to  which 
the  application  of  the  distinction  is  difficult;  and  an  attempt 
at  more  precise  definition  is  liable  to  involve  us  in  meta- 
physical discussions  such  as  the  logician  should  if  possible 
avoid.  The  first  question  that  arises  is  what  precisely  is 
meant  by  the  word  thing,  when  it  is  said  that  a  concrete 
name  is  the  name  of  a  thing.  By  a  thing,  we  may  mean 
anything  that  exists ;  but  we  cannot  mean  that  here,  since 
"attributes"  exist,  and  the  distinction  between  concrete 
and  abstract  names  would  vanish.  Again,  by  a  thing  we 
may  mean  a  substance ;  but  substances  are  contrasted  with 
feelings  as  well  as  with  attributes,  and  this  threefold  division 
would  make  names  of  feelings  neither  abstract  nor  concrete, 
which  can  hardly  be  intended.  With  regard  to  the  proper 
place  of  names  of  states  of  consciousness  it  would  be 
generally  agreed  to  call  them  concrete.  Thus,  while  seji- 
sibility,  the  faculty  of  experiencing  sensation,  is  an  abstract 
name,  the  name  of  a  sensation  itself  should  be  regarded 
as  concrete,  being  the  name  of  something  which  possesses 
attributes,  for  example,  of  being  pleasurable  or  painful,  of 
being  a  sensation  of  touch  or  one  of  hearing.  But  here 
again  a  difficulty  arises,  since,  as  pointed  out  by  iMill,  in 
many  cases  "  feelings  have  no  other  name  than  that  of  the 
attribute  which  is  grounded  on  them."  For  example,  by 
colour  we  may  mean  sensations  of  blue,  red,  green,  &c., 
or  we  may  mean  the  attribute  which  all  coloured  objects 
possess  in  common.  In  the  former  case,  colour  is  a  con- 
crete name,  in  the  latter  an  abstract  name.  Sound,  again, 
is  concrete,  in  so  far  as  it  is  the  name  of  a  sensation, 
e.g.^  "  the  same  sound  is  in  my  ears  which  in  those  days 
I  heard";  but  in  the  following  cases,  it  should  rather  be 


CHAP.  I.] 


TERMS. 


regarded  as  abstract, — "  a  tale  full  of  sound  and  fury,"  "  a 
name  harsh  in  sound." 

The  matter  is  still  further  complicated  if  Mill's  view  is 
taken,  and  attributes  are  analysed  into  sensations,  "the 
distinction  which  we  verbally  make  between  the  properties  of 
things  and  the  sensations  we  receive  from  them,  originating 
in  the  convenience  of  discourse  rather  than  in  the  nature 
of  what  is  signified  by  the  terms."  For  logical  purposes 
however  we  certainly  need  not  pursue  the  analysis  so  far 
as  this. 

But  still  another  difficulty  arises  from  the  fact  that  we 
sometimes  speak  of  attributes  themselves  as  having  attri- 
butes; and  so  far  as  this  is  permissible,  we  must  agree  with 
Professor  Jevons  that  "  abstractness  becomes  a  question  of 
degree.''  It  may  be  said  that  civilization  is  abstract  regarded 
as  an  attribute  of  a  given  state  of  society,  but  that  it  be- 
comes concrete  regarded  as  itself  possessing  the  attribute 
of  progressiveness  or  the  attribute  of  stationariness\ 

Besides  all  the  above,  we  have  to  notice  that  terms 
originally  abstract  are  very  liable  to  come  to  be  used  as 
concrete,  and  this  may  create  further  confusion.  Thus, 
Professor  Jevons  remarks, — ''Relation  properly  is  the  abstract 
name  for  the  position  of  two  people  or  things  to  each  other, 
and  those  people  are  properly  called  relatives.  But  we 
constantly  speak  now  of  relations,  meaning  the  persons 
themselves;  and  when  we  want  to  indicate  the  abstract 
relation  they  have  to  each  other  we  have  to  invent  a  new 
abstract  name  relationship.     Nation  has  long  been  a  con- 

^  It  does  not  however  follow  that  we  should  regard  the  name  of  a 
complex  attribute  as  therefore  concrete.  Civilization  regarded  as 
possessing  the  attribute  of  stationariness  may  be  considered  concrete, 
while  stationary  civilization  regarded  as  the  attribute  of  a  given  state  cf 
society  may  still  be  considered  abstract. 


lO 


TERMS. 


[part  I. 


Crete  term,  though  from  its  form  it  was  probably  abstract 
at  first ;  but  so  far  does  the  abuse  of  language  now  go, 
especially  in  newspaper  writing,  that  we  hear  of  a  nationality, 
meaning  a  nation,  although  of  course  if  nation  is  the  con- 
crete, nationality  ought  to  be  the  abstract,  meaning  the 
quality  of  being  a  nation.  Similarly,  action,  intuition,  ex- 
tension, conception,  and  a  multitude  of  other  properly  abstract 
names,  are  used  confusedly  for  the  corresponding  concrete, 
namely,  act^  intent,  extent,  concept,  &c."  (^Elementary  Lessons 
in  Logic,  pp.  21,  22). 

The  outcome  of  the  whole  discussion  seems  to  be  that  if 
we  are  asked  whether  a  given  name  is  abstract  or  concrete, 
we  frequently  cannot  give  an  absolute  answer,  but  have  to 
distinguish  between  different  cases.  Given  any  two  terms 
however  which  are  connected  together,  we  can  undertake 
to  say  which  of  them,  if  either,  is  abstract  in  relation  to 
the  other. 

7.  I  low  would  you  apply  the  distinction  between 
abstract    and    concrete    names    to    the    following: — 
life,  fate,  logic,   time,  fault,  generosity,   the  habit  of 
talking  loudly  ? 

8.  So  far  as  you  can,  name  the  concrete  terms 
corresponding  to  such  of  the  following  as  you  regard 
as  abstract,  and  the  abstract  terms  corresponding  to 
such  of  them  as  you  regard  as  concrete: — antithesis, 
Socrates,  attempt,  equation,  yelloiu,  ricJiness,  resentment, 
temper,  angel,  charity,  bounty,  compassion,  mei'cy. 

9.  Can    the    distinction    between    singular  and 
general  be  applied  to  abstract  names  .'* 

This  question  is  sometimes  answered  as  follows  : — Most 
abstract  names  are   general,  because   they  are   names   of 


XTHAP.  I.] 


TERMS. 


II 


attributes  which  are  found  in  different  objects.  Deity, 
however,  to  the  monotheist,  may  be  given  as  an  example 
of  a  singular  abstract,  since  it  is  the  name  of  an  attribute 
which  can  be  affirmed  of  God  only. 

This  criterion  would  make  the  corresponding  abstract 
of  every  general  concrete  name,  general,  and  of  every  singular 
concrete  name,  singular;  but  it  is  evidently  based  on  a 
fundamental  confusion.  By  an  abstract  name  we  mean  the 
name  of  an  attribute  considered  apart  from  the  things  pos- 
sessing that  attribute ;  and  the  attribute  is  to  be  regarded 
as  one  and  the  same  whether  it  is  possessed  by  one  thing 
only,  or  by  an  indefinite  number  of  things. 

Mill   takes   another  ground   of  distinction.     He   says, 
"  Some  abstract  names  are  certainly  general.     I  mean  those 
which  are  names  not  of  one  single  and  definite  attribute, 
but  of  a  class  of  attributes.     Such  is  the  word  colour^  which 
is  a  name  common  to  whiteness,  redness,  &c.    Such  is  even 
the  word  whiteness,  in  respect  of  the  various   shades   of 
whiteness  to  which  it  is  applied  in  common;    the  word 
magnitude,  in  respect  of  the  various  degrees  of  magnitude 
and  the  various  dimensions  of  space ;  the  word  weight,  in 
respect  of  the  various  degrees  of  weight.     Such  also  is  the 
word  attribute  itself,  the  common  name  of  all  particular 
attributes.     But  when  only  one  attribute,  neither  variable  in 
degree  nor  in  kind,  is  designated  by  the  name ;  as  visible- 
ness;  tangibleness ;  equality;    squareness;  milk-whiteness; 
then  the  name  can  hardly  be  considered  general;  for  though 
it  denotes  an  attribute  of  many  different  objects,  the  attri- 
bute itself  is  always  conceived  as  one  not  many"  {Logic,  i. 
p.  30).    I  should  doubt  if  any  attribute  can,  strictly  speaking, 
be  conceived  as  7na?iy.     An  attribute  in  itself  is  one  and 
indivisible,  and  does  not  admit  of  numerical  distinction. 
When  we  begin  to  distinguish  kinds  and  differences,  which 


12 


TERMS. 


[part  I. 


we  can  only  do  by  the  addition  of  other  attributes,  the  name 
would  appear  to  begin  to  partake  of  the  concrete  character. 
I  should  therefore  doubt  the  propriety  of  saying  that  some 
abstract  names  are  certainly  general.  It  would  be  more 
appropriate  to  call  all  strictly  abstract  names  singular. 
A  still  more  satisfactory  solution  however  is  to  consider 
the  distinction  of  general  and  singular  as  not  applying  to 
abstract  names  at  all.  Mill  himself  indicates  this  view, 
remarking  that,  "to  avoid  needless  logomachies,  the  best 
course  would  probably  be  to  consider  these  names  as  neither 
general  nor  individual,  and  to  place  them  in  a  class  apart" 
{Logic,  I.  p.  30). 

10.  Do  abstract  terms  admit  of  being  put  in  the 
plural  number }  Distinguish  between  the  terms  which 
are  abstract  and  concrete  in  the  following  list,  and  at 
the  same  time  indicate  which  can  m  your  opinion 
be  used  in  the  plural:— <f^/c;//r,  redness,  weight,  value, 
quinine,  equation,  heat,  zuannth,  hotness,  solitude,  zvhite- 
ness,  paper,  space,  gold.  [c.] 


CHAPTER  II. 


CONNOTATION   AND  DENOTATION. 


11.     The  Connotation  and  Denotation  of  Terms. 

Every  concrete  general  name  is  the  name  of  a  class,  real 
or  imaginary :  by  its  co?inotation  we  mean  the  attributes  on 
account  of  which  we  place  any  individual  in  the  class  or 
call  it  by  the  name ;  by  its  denotation  we  mean  the  in- 
dividuals which  possess  these  attributes,  and  which  are 
therefore  placed  in  the  class  and  called  by  the  name.  The 
terms  intension  (or  comprehension),  and  extension  are  also  used 
as  equivalent  to  connotation  and  denotation  respectively. 
Strictly  speaking  these  terms  belong  to  the  Conceptualist 
Logic,  and  should  be  applied  to  concepts  rather  than  to 
names. 

Thus,  the  connotation  of  *' plane  triangle"  is  given 
when  it  is  defined  as  a  plane  figure  contained  by  three 
straight  lines;  under  its  denotation  are  included  all  plane 
figures  fulfilling  this  condition.  The  connotation  of  "man" 
consists  of  those  attributes,  whatever  they  may  be,  which 
we  regard  as  essential  to  the  class  man,  i.e.,  in  the  absence 
of  any  one  of  which  we  should  refuse  to  call  any  individual 


H 


TERMS. 


[part  I. 


by  the  name;   its  denotation  is  made  up  of  all  the  indi- 
viduals actually  possessing  these  attributes. 

12.  Mill's  use  of  the  term  connotativc  compared 
with  that  of  other  writers. 

(i)  "A  non-connotative  term  is  one  which  signifies  a 
subject  only,  or  an  attribute  only.  A  connotative  term  is 
one  which  denotes  a  subject,  and  implies  an  attribute" 
(Mill,  Logic,  I.  p.  31).  According  to  this  definition,  a  con- 
notative name  must  possess  both  connotation  and  denotation. 

The  following  kinds  of  names  are  connotative  in  Mill's 
sense  : — (i)  All  concrete  general  names.  (2)  Some  singular 
names.  For  example,  "city"  is  a  general  name,  and  as 
such  no  one  would  deny  it  to  be  connotative.  Now  if  we 
say  "  the  largest  city  in  the  world  ",  we  have  individualised 
the  name,  but  it  does  not  thereby  cease  to  be  connotative. 
Proper  names  are,  however,  according  to  Mill,  not  conno- 
tative, since  they  merely  denote  a  subject  and  do  not  imply 
any  attributes.  To  this  point,  which  is  a  disputed  one,  we 
must  return.  Bain  {Logic,  Deduction,  p.  49)  implies  that 
only  general  names  are  connotative;  but  this  can  hardly 
have  been  intended.  (3)  Most  abstract  names  are  non- 
connotative,  since  they  merely  signify  an  attribute  and  do 
not  denote  a  subject.  Mill  however  maintains  that  some 
abstract  names  are  connotative,  namely,  the  names  of  attri- 
butes that  may  have  attributes  ascribed  to  them.  To  this 
point  also  we  must  return. 

(ii).  The  use  of  the  word  "  connotative"  does  not  seem 
to  have  been  quite  fixed  with  the  schoolmen.  Mansel  {Aid- 
rich,  p.  17),  while  admitting  that  there  was  some  license  in 
the  use  of  the  word,  gives  the  following  account  on  the 
authority  of  Occam.  With  the  schoolmen,  a  connotative 
term  was  one  that  **  primarily  signified  an  attribute,  second- 


CHAP.  II.] 


TERMS. 


15 


arily  a  subject ; "  (and  it  was  said  to  coimote  or  signify 
secondarily  the  subject).  Thus  **  white"  was  regarded  as  con- 
notative, whilst  the  original  substances  or  attributes,  as  "man" 
or  *'  whiteness  "  were  called  absolute;  the  former  signifying 
primarily  a  subject,  the  latter  not  signifying  a  subject  at 
all.  Only  adjectives  and  participles  therefore  (words  called 
by  Professor  Fowler  "attributives")  are  connotative  in  this 
sense. 

Mill  {Logic,  I.  p.  42,  note)  says  that  the  schoolmen  used 
it  in  his  own  sense,  though  some  of  their  expressions  are 
vague.  He  quotes  James  Mill  as  using  it  more  nearly  in 
the  sense  ascribed  by  Mansel  to  the  schoolmen. 

(iii)  Professor  Fowler  uses  the  term  connotative  in  a 
sense  different  from  that  of  Mill.  "A  term  may  be  said 
to  denote  or  designate  individuals  or  groups  of  individuals, 
to  cojinote  or  mean  attributes  or  groups  of  attributes."  In 
this  sense,  general  names  are  both  connotative  and  deno- 
tative; abstract  names  are  connotative  but  not  denotative', 
(whereas,  according  to  Mill,  they  are  generally  speaking 
denotative  but  not  connotative).  This  use  of  the  term 
avoids  some  difficulties,  and  I  am  inclined  to  regard  it  as 
preferable  to  Mijl's.  Indeed  Mill  himself  seems  to  suggest 
it  in  one  place.  He  says  that  James  Mill  "  describes 
abstract  names  as  being  properly  concrete  names  with 
their  connotation  dropped  :  whereas,  in  his  own  view,  it 
is  the  d^fc'notation  which  would  be  said  to  be  dropped,  what 
was  previously  connoted  becoming  the  whole  signification  " 
(Logic,  I.  p.  42  note). 

As  far  as  we  can  I  think  we  should  speak  merely  of  the 
"  denotation  "  and  "  connotation  "  of  names,  rather  than  of 
^'  denotative  "  and  "  connotative  names  ". 


^  Fowler,  Deductive  Logic,  p.  19. 


i6 


TERMS. 


[part  I. 


13.  Is  every  property  possessed  by  a  class  con- 
noted by  the  class-name  ? 

Unfortunately  we  do  not  find  complete  agreement  among 
logicians  with  regard  to  the  answer  that  should  be  given  to 
this  question ;  and  I  am  inclined  to  think  that  in  discussing 
points  connected  with  "connotation"  writers  sometimes 
misunderstand  each  other,  because  they  do  not  apprehend 
that  there  is  fundamental  disagreement  between  them  upon 

this  point. 

I  will  first  give  Mill's  answer  to  the  question,  an  answer 
with  which  I  should  myself  concur. 

By  the  connotation  of  a  class-name  he  does  not  mean 
all  the  properties  that  may  be  possessed  in  common  by  the 
class,  but  only  those  on  account  of  the  possession  of  which 
any  individual  is  placed  in  the  class,  or  called  by  the  name. 
In  other  words,  we  include  in  the  connotation  of  a  class- 
name  only  those  attributes  upon  which  the  classification  is 
founded,  and  in  the  absence  of  any  of  which  we  should  not 
regard  the  name  as  applicable.  For  example,  although  all 
equilateral  triangles  are  equiangular  we  should  not  include 
equiangularity  in  the  connotation  of  equilateral  triangle; 
although  all  kangaroos  may  happen  to  h^  Australhvi  kan- 
garoos, this  is  not  part  of  what  we  mean  to  imply  when 
we  use  the  name,— an  animal  subsequently  found  in  the 
interior  of  New  Guinea,  but  otherwise  possessing  all  the 
properties  of  kangaroos  would  not  have  the  name  kangaroo 
denied  to  it;  although  all  ruminant  animals  are  cloven- 
hoofed,  we  cannot  regard  cloven-hoofed  as  part  of  the 
mcanin^:  of  ruminant,  and  we  may  say  with  IMill  that  were 
an  animal  to  be  discovered  which  chews  the  cud,  but  has 
its  feet  undivided,  it  would  certainly  still  be  called  ru- 
minant. 

The  above  meaning  of  connotation  is  that  to  which  in 


CHAP.  II.] 


TERMS. 


17 


my  opinion  we  should  strictly  adhere.  It  is  of  course  open 
to  any  one  to  say  that  he  will  include  in  the  connotation  of 
a  class  name  all  the  properties  possessed  in  common  by  all 
members  of  the  class ;  but  this  is  simply  to  use  the  term  ifi 
a  different  sense.  It  is  used  in  this  sense  by  a  writer  in  a 
recent  number  of  Mitid.  "  On  the  connotative  side  a  name 
means,  to  us,  all  those  qualities  common  to  the  class  named 
with  which  we  are  acquainted; — all  those  properties  that  are 
said  to  be  *  involved  in  our  idea '  of  the  thing  named. 
These  are  the  properties  that  we  ascribe  to  an  object  when 
we  call  it  by  the  name.  But,  just  as  the  word  '  man,'  for 
example,  denotes  every  creature,  or  class  of  creatures  having 
the  attributes  of  humanity,  whether  we  know  him  or  not, 
so  does  the  word  properly  connote  the  whole  of  the  pro- 
perties common  to  the  class,  whetlier  we  know  them  or  not. 
Many  of  the  facts,  known  to  physiologists  and  anatomists 
about  the  constitution  of  man's  brain,  for  example,  are  not 
involved  in  most  men's  idea  of  the  brain :  the  possession  of 
a  brain  precisely  so  constituted  does  not,  therefore,  form 
any  part  of  their  meaning  of  the  word  'man.'  Yet  surely  this 
is  properly  connoted  by  the  word"  (E.  C.  Benecke,  in  Mind, 
1 88 1,  p.  532).  Professor  Jevons  also  uses  the  term  in  the 
same  sense.  "  A  term  taken  in  intent  (connotation)  has  for 
its  meaning  the  whole  infinite  series  of  qualities  and  circum- 
stances which  a  thing  possesses.  Of  these  qualities  or  cir- 
cumstances some  may  be  known  and  form  the  description 
or  definition  of  the  meaning;  the  infinite  remainder  are 
unknown"  {Pure  Logic,  p.  4).  Professor  Bain  appears  to 
use  the  term  in  an  intermediate  sense,  including  in  the 
connotation  of  a  class-name  not  all  the  attributes  common 
to  the  class  but  all  the  independent  attributes,  that  is,  all 
that  cannot  be  derived  or  inferred  from  others. 

It  ought  to  be  made  very  clear  in  any  discussion  con- 
K.  L.  2 


i8  TERMS.  [PART  I. 

cerning  the  connotation  of  names  in  which  of  these  several 
senses  we  are  using  the  term  "  connotation"  itself. 

It  may  be  said  that  to  use  the  term  in  Mill's  sense,  and 
to  make  connotation  depend  on  what  is  intended  to  be 
implied  by  the  mere  use  of  the  name,  is  to  make  it  vary 
with  every  different  speaker.  By  the  same  name  two  people 
may  mean  to  imply  different  things,  that  is,  the  attributes 
they  would  include  in  the  connotation  of  the  name  would 
be  different;  and  not  unfrequently  some  of  us  may  be 
unable  to  say  precisely  what  is  the  meaning  that  we  our- 
selves attach  to  the  words  we  use.  This  is  a  fact  which  it 
is  most  important  to  recognise.  But  for  the  purposes  of 
formal  logic  we  may  assume  that  every  name  has  a  fixed 
and  definite  connotation.  The  object  of  the  definition  of 
names  already  in  use  is  just  to  give  this ;  and  in  the  case 
of  an  ideal  language  properly  employed  every  name  would 
have  the  same  fixed  and  precise  meaning  for  everybody. 

14.  Arc  proper  names  connotative  or  non- 
connotative  ? 

On  the  question  here  raised  Mill  speaks  decisively, — 
"The  only  names  of  objects  which  connote  nothing  are 
proper  names;  and  these  have,  strictly  speaking,  no  signifi- 
cation" {Lo^ic,  I.  p.  36);  and  most  logicians  are  in  agree- 
ment with  him.  An  opposite  view  is  however  taken  by 
Jevons,  and  some  others  {e.g.,  F.  H.  Bradley,  T.  Shedden). 

In  one  or  two  places  I  am  inclined  to  think  that  Jevons 
tends  somewhat  to  obscure  the  point  at  issue.  Thus  with 
reference  to  Mill  he  says, — "Logicians  have  erroneously 
asserted,  as  it  seems  to  me,  that  singular  terms  are  devoid 
of  meaning  in  intension,  the  fact  being  that  they  exceed  all 
other  terms  in  tliat  kind  of  meaning"  {Principles  of  Science  ^ 
I-  PP-  32,  2>2>i  with  a  reference  to  Mill  in  the  foot-note). 


CHAP.  II.] 


TERMS. 


19 


But  IMill  distinctly  says  that  some  singular  names  are 
connotative,  e.g.y  the  sun,  the  first  emperor  of  Rome  {Logic, 
I.  pp.  34,  5).  Again,  Jevons  says, — "There  would  be  an 
impossible  breach  of  continuity  in  supposing  that  after 
narrowing  the  extension  of  *  thing '  successively  down  to 
animal,  vertebrate,  mammalian,  man,  Englishman,  educated 
at  Cambridge,  mathematician,  great  logician,  and  so  forth, 
thus  increasing  the  intension  all  the  time,  the  single  remain- 
ing step  of  adding  Augustus  de  Morgan,  Professor  in  Uni- 
versity College,  London,  could  remove  all  the  connotation, 
instead  of  increasing  it  to  the  utmost  point"  {Studies  in 
Deductive  Logic,  pp.  2,  3).  But  every  one  would  allow  that 
we  may  narrow  down  the  extension  of  a  term  till  it  becomes 
individualised  without  destroying  its  intension  or  connota- 
tion ;  "the  present  Professor  of  Pure  Mathematics  in  Uni- 
versity College,  London  "  is  a  singular  term, — we  cannot 
diminish  the  extension  any  further, — but  it  is  certainly 
connotative. 

We  must  then  clearly  understand  that  the  only  contro- 
versy is  with  regard  to  what  are  strictly /r^/^r  names.  Even 
yet  there  is  a  possible  source  of  ambiguity  that  should  be 
cleared  up.  If  by  the  connotation  of  a  name  we  mean  all 
the  attributes  possessed  by  the  individuals  denoted  by  the 
name,  or  even  all  the  independent  attributes.  Professor 
Jevons's  view  may  be  correct.  This  does  appear  to  be 
what  Jevons  himself  means,  but  it  is  distinctly  7iot  what 
Mill  means, — he  means  only  those  attributes  which  are 
implied  by  the  name  itself.  Jevons  puts  his  case  as 
follows  : — "Any  proper  name,  such  as  John  Smith,  is  almost 
without  meaning  until  we  know  the  John  Smith  in  question. 
It  is  true  that  the  name  alone  connotes  the  fact  that  he  is 
a  Teuton,  and  is  a  male ;  but,  so  soon  as  we  know  the 
exact  individual  it  denotes,  the  name  surely  implies,  also, 

2 — 2 


20 


TERMS. 


[part  I. 


the  peculiar  features,  form,  and  character,  of  that  individual. 
In  fact,  as  it  is  only  by  the  peculiar  qualities,  features,  or 
circumstances  of  a  thing,  that  we  can  ever  recognise  it,  no 
name  could  have  any  fixed  meaning  unless  we  attached  to 
it,  mentally  at  least,  such  a  definition  of  the  kind  of  thing 
denoted  by  it,  that  we  should  know  whether  any  given 
thing  was  denoted  by  it  or  not.  If  the  name  John  Smith 
does  not  suggest  to  my  mind  the  qualities  of  John  Smith, 
how  shall  I  know  him  when  I  meet  him  ?  for  he  certainly 
does  not  bear  his  name  written  upon  his  brow  "  {Elcmeiitivy 
Lessons  in  Logic,  p.  43).  A  wrong  criterion  of  connotation 
in  Mill's  sense  is  here  taken.  The  connotation  of  a  name 
is  not  the  quality  or  qualities  by  which  I  or  any  one  else 
may  happen  to  recognise  the  class  which  it  denotes.  For 
example,  I  may  recognise  an  Englishman  abroad  by  the  cut 
of  his  clothes,  or  a  Frenchman  by  his  pronunciation,  or  a 
proctor  by  his  bands,  or  a  barrister  by  his  wig ;  but  I  do 
not  7nean  any  of  these  things  by  these  names,  nor  do  they 
(in  Mill's  sense)  form  any  part  of  the  connotation  of  the 
names.  Compare  two  such  names  as  "John  Duke  Coleridge" 
and  "the  Lord  Chief  Justice  of  England."  They  denote 
the  same  individual,  and  I  should  recognise  John  Duke 
Coleridge,  and  the  Lord  Chief  Justice  of  England  by  the 
same  attributes ;  but  the  names  are  not  equivalent, — the  one 
is  given  as  a  mere  mark  of  a  certain  individual  to  distinguish 
him  from  others,  and  it  has  no  further  signification  ;  the 
other  is  given  on  account  of  the  performance  of  certain 
functions,  which  ceasing  the  name  would  cease  to  apply. 
Surely  there  is  a  distinction  here,  and  one  which  it  is 
important  that  we  should  not  overlook. 

Nor  is  it  true  that  such  a  name  as  "John  Smith*' 
connotes  "Teuton,  male,  &c."  John  Smith  might  be  a 
race-horse,  or  a  negro,  or  the  pseudonym  of  a  woman,  as  in 


CHAP.  II.] 


TERMS. 


21 


the  case  of  George  Eliot.  In  none  of  these  cases  could  a 
name  be  said  to  be  misapplied  as  it  would  if  a  horse  were 
called  a  man,  or  a  negro  a  Teuton,  or  a  woman  a 
male. 

But  it  may  fairly  be  said  that  in  a  certain  sense  many 
proper  names  do  suggest  something,  that  at  any  rate  they 
were  chosen  in  the  first  instance  for  a  special  reason.  For 
example,  Strongi'th'arm,  Smith,  Jungfrau.  Such  names 
however  even  if  in  a  certain  sense  connotative  when  first 
imposed  soon  cease  to  be  connotative  in  the  way  in  which 
other  names  are  connotative.  Their  application  is  in  no 
way  dependent  on  the  continuance  of  the  attribute  with 
reference  to  which  they  were  originally  given.  As  Mill  puts 
it,  "///^  7iame  once  given  is  independent  of  the  reason J^  Thus, 
a  man  may  in  his  youth  have  been  strong,  but  we  should 
not  continue  to  call  him  strong  when  he  is  in  his  dotage ; 
whilst  the  name  Strongi'th'arm  once  given  would  not  be 
taken  from  him.  The  name  "Smith"  may  in  the  first 
instance  have  been  given  because  a  man  plied  a  certain 
handicraft,  but  he  would  still  be  called  by  the  same  name 
if  he  changed  his  trade,  and  his  descendants  continue  to  be 
called  Smiths  whatever  their  occupations  may  be.  Nor  can 
it  be  said  that  the  name  necessarily  implies  ancestors  of 
the  same  name. 

Proper  names  of  course  become  connotative  when  they 
are  used  to  designate  a  certain  type  of  person  ;  for  example, 
a  Diogenes,  a  Thomas,  a  Don  Quixote,  a  Paul  Pry,  a 
Benedick,  a  Socrates.  But,  when  so  used,  such  names 
have  really  ceased  to  be  proper  names  at  all;  and  they 
have  come  to  possess  all  the  characters  of  general  names. 


15.     Discuss  the  question  whether  the  following 
terms    are   respectively  connotative  or  non-connota- 


22 


TERMS. 


[part  I. 


tive  : — Westminster   Abbey,   the   Mikado   of  Japan, 
Barmouth.  [L.] 

16.  Enquire  whether  the  following  names  are 
respectively  connotative  or  non-connotative: — Caesar, 
Czar,  Lord  Beaconsfield,  the  highest  mountain  in 
Europe,  Mont  Blanc,  the  Weisshorn,  Greenland,  the 
Claimant,  the  pole  star,  Homer,  a  Daniel  come  to 
judgment. 

17.  Can  any  abstract  names  possess  both  deno- 
tation and  connotation  .-* 

In  Fowler's  use  of  the  term  all  abstract  names  are  con- 
notative, that  is,  they  at  once  suggest  or  imply  attributes ; 
while  none  are  denotative,  that  is,  they  do  not  denote 
individuals  or  groups  of  individuals.  Professor  Fowler 
himself  admits  that  it  sounds  paradoxical  to  say  that 
abstract  names  are  not  denotative,  but  he  is  of  opinion  that 
the  employment  of  the  expressions  in  his  sense  would 
simplify  the  statement  and  explanation  of  many  logical 
difficulties.  I  am  inclined  to  think  that  the  present  is  a 
case  in  point. 

Mill  holds  that  while  most  abstract  names  are  non-con- 
notative, still  ''even  abstract  names,  though  the  names  only 
of  attributes,  may  in  some  instances  be  justly  considered  as 
connotative  ;  for  attributes  themselves  may  have  attributes 
ascribed  to  them ;  and  a  word  which  denotes  attributes  may 
connote  an  attribute  of  those  attributes"  {LogiCy  i.  p.  2>?t)-  I 
have  some  difficulty  in  interpreting  this  passage.  Suppose 
that  we  have  a  connotative  abstract  name  denoting  the  attri- 
bute A  and  connoting  the  attribute  B ;  now  a  connotative 
name  is  always  defined  by  means  of  its  connotation,  and  we 
shall  therefore  define  our  term  by  saying  that  it  connotes  B 


CHAP.  IT.] 


TERMS. 


23 


without  any  reference  whatever  to  A.  What  then  w4ll  dis- 
tinguish it  from  the  concrete  term  denoting  whatever  pos- 
sesses B  ?  The  solution  of  the  difficulty  seems  to  be  that 
when  we  talk  of  one  attribute  having  another  ascribed  to  it, 
the  term  denoting  it  becomes  concrete  rather  than  abstract. 
Comparing  Mill's  definitions  of  an  abstract  name  and  of  a 
connotative  name,  I  fail  to  understand  how  the  same  name 
can  be  both  \ 

18.  Explain  and  discuss  the  statement : — "  In  a 
series  of  common  terms  arranged  in  regular  sub- 
ordination to  one  another,  the  denotation  and  con- 
notation vary  inversely." 

19.  Explain  the  following  statements : — 

{a)  If  a  term  be  abstract,  its  denotation  is  the 
same  as  the  connotation  of  the  corresponding  con- 
crete ? 

{b)  Of  the  denotation  and  connotation  of  a  term, 
one  may,  both  cannot,  be  arbitrary. 

{c)  Names  with  indeterminate  connotation  are  not 
to  be  confounded  with  names  which  have  more  than 
one  connotation. 

20.  Verbal  and  Real  Propositions. 

A  Verbal  Proposition  is  one  in  which  the  connotation 

^  Mr  Killick  in  his  Handbook  of  Milts  Logic  makes  Mill  include  in 
the  class  of  connotative  names  such  abstract  names  as  are  the  names  of 
groups  of  attributes  {e.g.,  /uimanity).  I  do  not  think  that  Mill  himself 
intended  this,  nor  do  I  think  that  the  view  is  a  correct  one  {i.e.,  accord- 
ing to  Mill's  own  usage  of  terms).  If  an  abstract  name  has  both  deno- 
tation and  connotation  because  it  is  the  name  of  a  group  of  attributes, 
on  what  principle  shall  we  distinguish  between  the  attributes  that  it 
denotes  and  those  that  it  connotes  ? 


24 


TERMS. 


[part  I. 


of  the  predicate  is  a  part  or  the  whole  of  the  connotation 
of  the  subject.  Bain  describes  the  verbal  proposition  as 
"the  notion  under  the  guise  of  the  proposition";  and  it  is 
certainly  convenient  to  discuss  verbal  propositions  in  con- 
nection with  the  connotation  of  names  or  the  intension  of 
concepts.  The  most  important  class  of  verbal  propositions 
are  definitions,  the  essential  function  of  which  is  to  analyse 
the  connotation  of  names'.  The  least  important  class  are 
absolutely  tautologous  or  identical  propositions,  e.g.,  all  A 
is  Aj  a  man  is  a  man. 

Real  Propositions,  on  the  other  hand,  "predicate  of  a 
thing  some  fiict  not  involved  in  the  signification  of  the 
name  by  which  the  proposition  speaks  of  it ;  some  attribute 
not  connoted  by  that  name." 

The  same  distinction  is  also  expressed  by  the  pairs  of 
terms,  analytic '^  and  synthetic,  explicative^  and  ampliative, 
essential^  and  accidental. 

^  Besides  propositions  giving  such  an  analysis  more  or  less  com- 
plete, the  following  classes  of  propositions  are  frequently  included  under 
the  head  of  verbal  propositions  :  where  the  subject  and  predicate  are 
both  proper  names,  e.g..,  Tully  is  Cicero;  where  they  are  dictionary 
synonyms,  c.g.^  wealth  is  riches,  a  story  is  a  tale,  charity  is  love. 

All  such  propositions  however  can  hardly  be  brought  under  the  head 
of  verbal  propositions  as  defined  in  the  text.  At  any  rate  if  we  have 
decided  that  a  proper  name  is  not  connotative,  it  is  clear  that  in  no 
proposition  having  a  proper  name  for  its  subject  can  the  predicate  be 
any  part  of  the  connotation  of  the  subject. 

To  include  these  classes  we  must  define  a  verbal  proposition  as  a 
proposition  which  is  wholly  concerned  with  the  meaning  or  application 
of  names,  a  real  proposition  as  one  which  is  concerned  with  things  or 
qualities. 

Even  with  these  definitions,  however,  while  it  is  a  verbal  proposition 
to  say  that  Tully  is  Cicero  (i.e.,  that  these  names  have  the  same  appli- 
cation), it  is  a  real  proposition  to  say  that  Tully  is  an  individual  who  is 
also  denoted  by  the  name  Cicero. 

2  It  should  be  carefully  observed  that  while  the  term  verbal  is  some- 


CHAP.  II.] 


TERMS. 


25 


21.    Which  of  the  following  propositions  should 
you  regard  as  Real,  and  why } 

Homer  wrote  the  Iliad, 

Instinct  is  untaught  ability. 

Instinct  is  hereditary  experience.  [c] 

"Homer  wrote  the  Iliad"  is  regarded  by  Bain  as  a 
verbal  predication.  "  We  know  nothing  about  Homer  except 
the  authorship  of  the  Iliad.  We  have  not  a  meaning  to 
attach  to  the  subject  of  the  proposition,  *  Homer',  apart 
from  the  predicate,  *  wrote  the  Iliad.'  The  affirmation  is 
nothing  more  than  that  the  author  of  the  Iliad  was  called 
Homer"  {Logic,  Dcdiictmi,  p.  67).  Taking  the  definition  of 
verbal  proposition  given  in  the  text,  and  holding  that  no 
proper  name  is  connotative,  this  view  must  clearly  be 
rejected.  If  however  by  a  verbal  proposition  we  mean  one 
that  relates  in  any  way  to  the  application  of  names,  (i.e., 
taking  the  definition  given  in  the  note),  there  may  be  some- 
thing to  say  for  it.  But  is  it  true  that  we  attach  nothing 
more  to  "Homer"  than  ''wrote  the  Iliad"?  Do  we  not, 
for  example,  attach  to  "Homer"  the  authorship  of  other 
poems,  and  also  an  individuality  ^  ?  If  it  is  the  fact  that 
the  Iliad  was  the  work  of  various   authors,  as  has  been 


times  stretched  so  as  to  include  such  a  proposition  as  "Tully  is  Cicero," 
this  is  never  the  case  with  the  terms  analytic,  explicative,  essential. 
These  terms  are  strictly  limited  to  propositions  which  give  no  informa- 
tion whatever  (even  with  regard  to  the  application  of  names)  to  any  one 
who  is  fully  acquainted  with  the  connotation  or  intension  of  the  subject 
term. 

*  I  do  not  of  course  mean  that  this  is  the  connotation  of  •'  Homer," 
for  I  hold  that  no  proper  names  are  connotative.  I  mean  that  Homer 
denotes  for  me  a  certain  individual  who  was  a  Greek,  who  lived  prior 
to  a  certain  date,  and  who  was  the  author  of  certain  poems  other  than 
the  Iliad. 


26 


TERMS. 


[part  I. 


asserted,  would  not  the  proposition  become  false?  Still,  we 
should  perhaps  admit  that  we  have  here  a  limiting  case. 
Some  light  may  be  thrown  on  the  point  thus  raised  by  an 
answer  once  sent  in  by  an  examinee:  "The  accepted 
opinion  is  that  the  Iliad  was  not  written  by  Homer,  but  by 
another  man  of  the  same  name." 

"Instinct  is  untaught  ability"  and  "Instinct  is  here- 
ditary experience"  may  be  regarded  as  verbal  and  real 
respectively. 

22.  Is  it  a  verbal  proposition  to  say  that  it  is 
hotter  in  summer  than  in  winter } 

Examine  the  following  statements:  A  free  in- 
stitution is  a  contradiction  in  terms ;  so  is  a  perfect 
creature.  L^-J 

23.  If  all  X  is  f,  and  some  x  is  5,  and  /  is  the 
name  of  those  ^'s  which  are  x;  is  it  a  verbal  pro- 
position to  say  that  all/  isj^  }  [v.] 

24.  Give  one  example  of  each  of  the  following, — 
(i)  a  collective  general  name,  (ii)  a  singular  abstract 
name,  (iii)  a  connotative  abstract  name,  (iv)  a  con- 
notative  singular  name  ;  or,  if  you  deny  the  possi- 
bility of  any  of  these  combinations,  state  clearly  your 
reasons. 


CHAPTER  III. 


POSITIVE  AND   NEGATIVE  NAMES.      RELATIVE  NAMES. 


25.     Positive  and  Negative  Terms. 

The  essential  distinction  between  positive  and  negative 
names  as  ordinarily  understood  may  be  expressed  as 
follows : — a  positive  name  implies  the  presence  of  certain 
definite  attributes;  a  negative  name  implies  the  absence 
of  one  or  other  of  certain  definite  attributes. 

"Every  name,"  as  remarked  by  De  Morgan,  "applies 
to  everything  positively  or  negatively";  for  example,  every- 
tliing  either  is  or  is  not  a  horse.  Every  name  tlien  divides 
all  things  in  the  universe  into  two  classes.  Of  one  of  these 
it  is  itself  the  name ;  and  a  corresponding  name  can  be 
framed  to  denote  the  other.  This  pair  of  names,  which 
between  them  denote  the  whole  universe,  are  respectively 
positive  and  negative.  But  which  is  which  ?  Which  is  the 
negative  name,  since  each  positively  denotes  a  certain  class 
of  objects?  The  distinction  lies  in  the  manner  in  which 
the  class  is  determined.  We  may  say  that  in  a  certain 
sense  a  strictly  negative  name  has  not  an  independent 
connotation  of  its  own ;  its  denotation  is  determined  by  the 
connotation  of  the  corresponding  positive  name.  It  denotes 
an  indefinite  and  unknown  class  outside  a  definite  and 
limited  class.     In  other  words,  by  means  of  its  connotation 


28 


TERMS. 


[part  1. 


we  first  mark  off  the  class  denoted  by  the  positive  name, 
and  then  the  negative  name  denotes  what  is  left.  The  fact 
that  its  denotation  is  thus  determined  is  the  distinctive 
characteristic  of  the  negative  name. 

We  have  here  supposed  that  between  them  the  positive 
name  and  the  corresponding  negative  name  exhaust  the 
whole  universe.  But  something  different  from  this  is  often 
meant  by  a  negative  name.  Thus  De  Morgan  considers 
that  parallel  and  alien  are  negative  names.  *'In  the 
formation  of  language,  a  great  many  names  are,  as  to  their 
original  signification,  of  a  purely  negative  character :  thus, 
parallels  are  only  lines  which  do  not  meet,  aliens  are  men 
who  are  not  Uritons  (/>.,  in  our  country)"  {Formal  Logic, 
p.  37).  But  these  names  clearly  have  not  the  thorough- 
going negative  character  that  I  have  just  been  ascribing  to 
negative  names.  The  difference  will  be  found  to  consist  in 
this,  that  in  the  sense  in  which  alien  is  a  negative  name, 
the  positive  and  negative  names  (Briton  and  alien)  do  not 
between  them  exhaust  the  entire  universe,  but  only  a  limited 
universe,  namely,  in  the  given  case,  that  constituted  by  the 
inhabitants  of  Great  Britain.  We  may  perhaps  distinguish 
between  names  absolutely  Jiegative,  where  the  reference  is  to 
the  entire  universe;  and  names  relatively  negative,  where 
the  reference  is  only  to  some  limited  universe. 

Now  it  will  be  seen  that  in  the  use  of  such  a  term  as 
not-white  there  is  a  possible  ambiguity;  we  must  decide 
whether  in  any  given  instance  the  name  is  to  be  regarded 
as  absolutely  or  only  as  relatively  negative.  ISIill  chooses 
the  former  alternative;  *' not-white,"  he  says,  ** denotes  all 
things  whatever  except  white  things."  De  Morgan  and 
Bain  however  consider  that  in  such  a  case  the  reference  is 
not  to  the  whole  universe  but  to  some  particular  universe 
only.     Thus,  in   contrasting  white  and   not-white  we   are 


CHAP.  III.] 


TERMS. 


29 


referring  solely  to  the  universe  of  colour;  not-7vhite  does 
not  include  everything  in  nature  except  white  things,  but 
only  things  that  are  black,  red,  green,  yellow,  &c.,  that  is, 
all  coloured  things  except  such  as  are  white '.  Whately  and 
Jevons  agree  with  Mill ;  and  from  a  logical  point  of  view 
I  think  they  are  right.  Or  rather  I  w^ould  say  that  two  such 
terms  as  S  and  not-^  must  between  them  exhaust  the 
universe  of  discourse^  whatever  that  may  be ;  and  we  must 
not  be  precluded  from  making  this,  if  we  care  to  do  so,  the 
entire  universe  of  existence.  That  is,  not-^  may  he  called 
upon  to  assume  the  absolutely  negative  character".  For  if 
we  are  unable  to  denote  by  not-.S'  all  things  whatsoever 
except  S,  it  is  difficult  to  see  in  what  way  we  shall  be  able 
to  denote  these  when  we  have  occasion  to  refer  to  them. 
On  the  other  hand,  we  must  also  be  empowered  to  indicate 
a  limitation  to  a  particular  universe  where  that  is  intended. 
By  not-5  then  referred  to  without  qualification  expressed  or 
implied  by  the  context  I  would  understand  the  absolute 
negative  of  S;  but  I  should  be  quite  prepared  to  find  a 
limitation  to  some  more  restricted  universe  in  any  particular 
instance. 

It  should  be  noted  that  in  the  case  of  a  limited  uni- 
verse it  is  sometimes  difficult  to  say  which  of  the  pair  of 
contrasted  names  is  really  to  be  regarded  as  the  negative 
name.  For  example,  De  Morgan  says  that  parallel  is  a 
negative  name,  since  parallel  lines  are  simply  lines  that  do 
not  meet.    But  we  might  also  define  them  as  lines  such  that 


^  Thus,  on  Bain's  view  it  would  be  incorrect  to  say  that  an  im- 
material entity  such  as  honesty  was  not-white. 

*  On  this  view,  "not-white"  might  be  used  to  denote  not  merely 
coloured  things  that  are  not  white,  but  also  things  that  are  not  coloured 
at  all.  It  would  for  example  be  correct  to  say  that  honesty  was  not- 
white. 


so 


TERMS. 


[part  I. 


if  another  line  be  drawn  cutting  them  both,  the  alternate 
angles  are  equal  to  one  another;  and  then  the  name  appears 
as  a  positive  name.  Similarly  in  the  universe  of  property, 
as  pointed  out  by  De  Morgan,  personal  and  real  are 
respectively  the  negatives  of  each  other ;  but  if  we  are  to 
call  one  positive  and  the  other  negative,  it  is  not  quite  clear 
which  should  be  which. 

For  a  suggestion  of  IMr  Monck's  as  to  the  definition  of 
negative  terms,  see  section  29. 

26.  Privative  Names. 

To  the  distinction  between  positive  and  negative  names, 
Mill  adds  a  class  of  names  called  privative.  "  A  privative 
name  is  ecjuivalent  in  its  signification  to  a  positive  and  a 
negative  name  taken  together;  being  the  name  of  some- 
thing which  has  once  had  a  particular  attribute,  or  for 
some  other  reason  might  have  been  expected  to  have  it,  but 
which  has  it  not.  Such  is  the  word  blind^  which  is  not 
equivalent  to  7iot  seeing,  or  to  not  capable  of  seeing,  for  it 
would  not,  except  by  a  poetical  or  rhetorical  figure,  be 
applied  to  stocks  and  stones"  {Logic,  i.  p.  44).  Perhaps 
also  idle,  which  Mill  gives  as  a  negative,  should  rather  be 
regarded  as  a  privative  term.  It  does  not  mean  merely 
"not-working,"  but  "not-working  where  there  is  the  capacity 
to  work."  We  should  hardly  speak  of  a  stone  as  being 
"idle." 

The  distinction  here  indicated  does  not  appear  to  be  of 
logical  importance. 

27.  How  far  is  it  true  that,  as  ordinarily  under- 
stood, negative  terms  have  a  definite  connotation, 
while  in  Logic  they  have  not }  So  far  as  it  is  true, 
how  would  you  explain  the  fact  ?  [w.] 


CHAP.  III.]  TERMS.  31 

28.    Coiiti'adictory  and  contrary  terms. 

A  positive  term  and  its  corresponding  negative  term  are 
called  contradictories.  A  pair  of  contradictory  terms  are  so 
related  that  between  them  they  exhaust  the  entire  universe 
to  which  reference  is  made,  whilst  in  that  universe  there  is 
no  mdividual  of  which  both  can  be  at  the  same  time 
affirmed.  The  nature  of  this  relation  is  expressed  in  the 
two  laws  of  Contradiction  and  Excluded  Middle.  Nothin- 
IS  at  the  same  time  both  X  and  not-X;  Everything  is  X 
or  not-X  For  the  application  of  the  above  to  complex 
terms,  see  Part  iv. 

The  contrary  of  a  term  is  usually  defined  as  the  term 
denoting  that  which  is  furthest  removed  from  it  in  some 
particular  universe ;  e.g.,  black  and  white,  wise  and  foolish. 
Two  contraries  may  in  some  cases  happen  to  make  up 
between  them  the  whole  of  the  universe  in  question,  e.g., 
Briton  and  alien;  but  this  is  not  necessary,  e.g.,  black  and 
white.  It  follows  that  although  two  contraries  cannot  both 
be  true  of  the  same  thing  at  the  same  time,  they  may  both 
be  false. 

The  above  may  be  called  the  material  contrary.  In  the 
case  of  complex  terms,  we  may  also  assign  a  formal  con- 
trary, as  is  shewn  in  Part  iv. 


29.  Illustrate  Mill's  statement  that  "  names  which 
are  positive  in  form  are  often  negative  \x\  reality,  and 
others  are  really  positive  though  their  form  is  neira- 
tive."  "^ 

The  fact  that  a  really  positive  term  is  sometimes  negative 
in  form  results  from  the  circumstance  that  the  negative  pre- 
fix IS  sometimes  given  to  the  contrary  of  a  term.  But  we 
have  seen  that  a  term  and  its  contrary  may  both  be  positive. 


32 


TERMS. 


[part  I. 


For  example,  pleasant  and  unpleasant;  ''the  word  un- 
pleasant, notwithstanding  its  negative  form,  does  not  con- 
note the  mere  absence  of  pleasantness,  but  a  less  degree  of 
what  is  signified  by  the  word  painfuL  which,  it  is  hardly 
necessary  to  say,  is  positive."  On  the  other  hand,  some 
names  positive  in  form  may  be  regarded  as  relatively  nega- 
tive, e.g.,  parallel,  alien.  I  do  not  however  think  that  an 
absolutely  negative  name  can  be  found  that  is  positive  in 

form. 

But  for  purposes  of  formal  logic  it  does  not  much 
concern  us  whether  any  given  term  is  positive  or  negative. 
What  the  formal  logician  is  really  concerned  with  is  the 
relation  between  contradictory  terms.  Not-^"  is  the  contra- 
dictory of  5,  and  5  is  the  contradictory  of  not-5,  whichever 
of  the  terms  may  be  more  strictly  the  positive  and  the 
negative  respectively. 

Mr  Monck,  in  his  valuable  Introduction  to  Logic,  p.  104, 
suggests  that  it  might  be  "  better  to  define  a  Negative  term 
as  a  term  negative  in  form,  (i.e.,  a  term  in  which  'non,' 
*un,'  'in,'  'mis,'  or  some  other  negative  particle  occurs)." 
In  my  opinion,  this  suggestion  miglit  without  disadvantage 
be  adopted. 

30.  Truth  applies,  it  is  said,  only  to  propositions. 
If,  then,  a  simple  term  is  not  capable  of  truth,  it 
must  be  false;  because  everything  must  be  cither  true 
or  false.     Solve  this  difficulty.  [L.] 

31.  "  For  every  positive  concrete  name  a  corre- 
sponding negative  one  might  be  framed."  Illustrate 
the  meaning  of  this  statement,  and  find  the  precise 
negatives  of  the  positive  terms  Man,  PJiysician,  Red, 


Thing, 


[L.] 


CHAP.  III.]  TERMS. 

32.     Relative  Names. 


33 


"  A  name  is  relative,  when,  over  and  above  the  object 
which  it  denotes,  it  implies  in  its  signification  the  existence 
of  another  object,  also  deriving  a  denomination  from  the 
same  fact  which  is  the  ground  of  the  first  name."  (Mill, 
Logic,  I.  p.  47-) 

Jevons  considers  that  all  terms  are  in  one  sense  relative. 
By  the  law  of  relativity,  consciousness  is  possible  only 
under  circumstances  of  change.  Every  term  therefore  im- 
plies its  negative  as  an  object  of  thought.  For  example, 
take  the  term  7nan.  It  is  an  ambiguous  term,  and  in  many 
of  its  meanings  it  is  strikingly  relative, — for  example,  as 
opposed  to  master,  to  officer,  to  woman,  to  wife,  to  boy. 
If  in  any  sense  it  is  absolute,  i.e.,  not  relative,  it  is  when 
opposed  to  not-man ;  but  even  in  this  case  it  may  be  said 
to  be  relative  to  not-man.  To  avoid  this  difficulty,  Jevons 
remarks,  "Logicians  have  been  content  to  consider  as  rela- 
tive terms  those  only  which  imply  some  peculiar  and  striking 
kind  of  relation  arising  from  position  in  time  or  space,  from 
connexion  of  cause  and  effect,  &c. ;  and  it  is  in  this  special 
sense  therefore  that  the  student  must  use  the  distinction." 

It  is  a  little  doubtful  however  whether  every  name  can 
be  said  to  imply  its  negative  /;/  its  signification.  Because  all 
things  are  relative  does  it  necessarily  follow  that  all  tcnns 
are  relative  ?  The  matter  is  of  no  great  importance,  and  at 
any  rate  the  difficulty  might  be  avoided  by  defining  a 
relative  term  as  one  which  implies  in  its  signification  the 
existence  of  another  object,  other  than  its  mere  negation. 

The  fact  or  facts  constituting  the  ground  of  both  correla- 
tive  names   is   called    the  fundamentuin    relationis.      For 
example,  in  the  case  of  partner,   the  fact  of  partnership ; 
in  the  case  of  husband  and  wife,  the  facts  which  constitute 
K.  L.  3 


34 


TERMS. 


[part  I. 


the  marriage  tie  ;  in  the  case  of  shepherd  and  sheep,  the 
acts  of  tending  and  watching  which  the  former  exercises 

over  the  latter. 

Sometimes  the  relation  which  each  correlative  bears  to 
the  other  is  the  same  ;  for  example,  in  the  case  of  partner, 
where  the  correlative  name  is  the  same  name  over  again. 
Sometimes  it  is  not  the  same ;  for  example,  father  and  son, 
husband  and  wife. 

33.  Describe  in  logical  phrase  the  character  of 
the  following  words :— man,  Peter,  humanity,  the  sun, 
post,  idle,  unpleasant,  daughter.  [c] 

In  dealing  with  any  term  for  logical  purposes,  we  must 
first  of  all  determine  whether  it  is  univocal,  that  is,  used  in 
one  definite  sense  only,  or  equivocal  (or  ambiguous),  that  is, 
used  in  more  senses  than  one.  In  the  latter  case,  we  may 
find  that  its  logical  characteristics  vary  according  to  the 
sense  in  which  it  is  used. 

34.  What  are  the  logical  characteristics  of  the 
terms -.—beauty,  immortal,  slave,  England,  Paradise, 
friendship,  law,  sovereign,  the  Times,  the  Arabian 
Nights,  George  Eliot,  Mrs  Grundy,  Vanity  Fair,  sleep, 
truth,  selfish,  ungenerous,  nobility,  treason  ? 


PART   II. 


PROPOSITIONS. 


CHAPTER  I. 


KINDS    OF   PROPOSITIONS.      THE   QUANTITY   AND    QUALITY 

OF   PROPOSITIONS. 

35.  Categorical,  Hypothetical  and  Disjunctive 
Propositions. 

For  logical  purposes,  a  Proposition  may  be  defined  as 
"a  sentence  indicative  or  assertory,"  (as  distinguished,  for 
example,  from  sentences  imperative  or  exclamatory);  in 
other  words,  a  proposition  is  a  sentence  making  an  affirma- 
tion or  denial,  as— All  Sv^  P ,  No  vicious  man  is  happy. 

A  proposition  Is  Categorical  if  the  affirmation  or  denial 
is  absolute,  as  in  the  above  examples.  It  is  Hypothetical 
if  made  under  a  condition,  as— If  y^  is  ^,  C\^  D\  Where 
ignorance  is  bliss,  'tis  folly  to  be  wise.  It  is  Disjutictive  if 
made  with  an  alternative,  as — Either  P  is  <2,  or  X  is  Y\ 
He  is  either  a  knave  or  a  fool\ 

^  It  should  be  observed  that  in  a  disjunctive  proposition  there  may 
be  two  distinct  subjects  as  in  the  first  of  the  above  examples,  or  only 
one  as  in  the  second.  Disjunctive  propositions  in  which  there  is  only 
one  distinct  subject  are  the  more  amenable  to  logical  treatment. 

3—2 


36 


PROPOSITIONS. 


[part  II. 


[The  above  threefold  division  is  adopted  by  Mansel. 
It  is  perhaps  more  usual  to  commence  with  a  twofold 
division,  the  second  member  of  which  is  again  subdivided, 
the  term  Hypothetical  being  employed  sometimes  in  a  wider 
and  sometimes  in  a  narrower  sense.  To  prevent  confusion, 
it  may  be  helpful  to  give  the  following  table  of  the  usage' 
of  one  or  two  modern  logicians  with  regard  to  this  division. 

Whately,  Mill  and  Bain  : — 


I.     Categorical. 


2.     Hypothetical,       / 

or  Compound,  Y'^  Conditional, 
or  Complex.      r^   Disjunctive. 

Hamilton  and  Thomson  : — 

1.  Categorical. 

2.  Conditionally^    Hypothetical. 

((2)   Disjunctive. 

Fowler  (following  Boethius) : — 

1 .  Categorical. 

2.  Conditional  ((i)   Conjunctive. 

or  Hypothetical.  ((2)   Disjunctive. 

Mansel,  as  I  have  already  remarked,  gives  at  once  a 
threefold  division. 

1.  Categorical. 

2.  Hypothetical  or  Conditional. 

3.  Disjunctive. 

He  states  his  reasons  for  his  own  choice  of  terms  as 
follows  :— "  Nothing  can  be  more  clumsy  than  the  employ- 
ment of  the  word  couditiotial  in  a  specific  sense,  while  its 
Greek  eciuivalent,  hypothetical,  is  used  generically.  In  Boe- 
thius, both  terms  are  properly  used  as  synonymous,  and 
generic ;  the  two  species  being  called  conjunctivi,  cortjuncii, 


CHAP.  I.] 


PROPOSITIONS. 


or  conncxi,  and  disjunctivi  or  disjunctt.  With  reference  to 
modern  usage,  however,  it  will  be  better  to  contract  the 
Greek  word  than  to  extend  the  Latin  one.  Hypotheticalin 
the  following  notes,  will  be  used  as  synonymous  with  con- 
ditiojiar  (ManseFs  edition  oi  Aldrich,  p.  103).] 

36.  A  logical  analysis  of  the  Categorical  Pro- 
position. 

In  logical  analysis,  the  categorical  proposition  always 
consists  of  three  parts,  namely,  two  terms  which  are  united 
by  means  of  a  copula. 

The  sjibjed  is  that  term  about  which  affirmation  or  denial 
IS  made ;  it  represents  some  notion  already  partially  deter- 
mined in  our  mind,  and  which  it  is  our  aim  further  to 
determine. 

The  predicate  is  that  term  which  is  affirmed  or  denied 
of  the  subject;  it  enables  us  further  to  determine  the 
subject,  i.e.,  to  enlarge  our  knowledge  with  regard  to  it. 

The  copula  is  the  link  of  connection  between  the  subject 
and  the  predicate,  and  consists  of  the  words  is  or  is  7iot 
according  as  we  affirm  or  deny  the  latter  of  the  former. 

In  attempting  to  apply  the  above  analysis  to  such  a 
proposition  as  ''All  that  love  virtue  love  angling,"  we  find 
that,  as  it  stands,  the  copula  is  not  separately  expressed. 
It  may  however  be  written, — 

subj.  cop.  pred. 

All  lovers  of  virtue  |  are  |  lovers  of  angling; 

and  in  this  form  the  three  different  elements  of  the  logical 
proposition  are  made  distinct.  This  analysis  should  always 
be  performed  in  the  case  of  any  proposition  that  may 
at  first  present  itself  in  an  abnormal  form.  A  difficulty 
that   may   sometimes   arise  in  discriminating  the  subject 


38 


PROPOSITIONS. 


[part  II. 


and   the   predicate  is   dealt  with  subsequently, — compare 
section  50. 

The  older  logicians  distinguished  propositions  secufidi 
adjacentisy  and  propositions  tertii  adjacentis.  In  the  former, 
the  copula  and  the  predicate  are  not  separated ;  e.g.,  The 
man  runs,  All  that  love  virtue  love  angling.  In  the  latter, 
the  copula  and  the  predicate  are  made  distinct ;  e.g..  The 
man  is  running,  All  lovers  of  virtue  are  lovers  of  angling. 
A  categorical  proposition,  therefore,  when  expressed  in 
exact  logical  form,  is  tertii  adjacentis. 

37.  Exponible^  copulative^  exclusive,  exceptive  pro- 
positions. 

Propositions  that  are  resolvable  into  more  propositions 
than  one  have  been  called  exponible,  in  consequence  of  their 
susceptibility  of  analysis.  Copulative  propositions  are  formed 
by  a  direct  combination  of  simple  propositions,  e.g.,  P  is 
both  Q  and  R  {i.e.,  F  is  Q,  F  is  F),  A  is  neither  B  nor 
C  {i.e.,  A  is  not  B,  A  is  not  C);  they  form  one  class 
of  exponibles.  Exclusive  propositions  contain  some  such 
word  as  "  only,"  thereby  limiting  the  predicate  to  the  sub- 
ject ;  e.g.,  Only  S  is  F.  This  may  be  resolved  into  S  is  F, 
and  F  is  S.  Propositions  of  this  kind  also  are  therefore 
exponibles.  Exceptive  propositions  limit  the  subject  by  such 
a  word  as  "unless"  or  "except";  e.g.,  A  is  X,  unless  it 
happens  to  be  B.  These  too  may  perhaps  be  regarded  as 
exponiblc  propositions. 

38.  The  Quantity  and  Quality  of  Propositions. 

The  Quality  of  a  proposition  is  determined  by  the 
copula,  being  ajjlrmative  or  negative  according  as  the  copula 
is  of  the  form  "is"  or  "is  not." 

Propositions  are  also  divided  into  universal  and  parii- 


CHAP.  I.]  PROPOSITIONS.  39 

cular,  according  as  the  affirmation  or  denial  is  made  of  the 
whole  or  only  of  a  part  of  the  subject.  This  division  of 
Propositions  is  said  to  be  according  to  their  Quantity. 

Combining  the  two  principles  of  division,  we  get  four 
fundamental  forms  of  propositions  : — 

(i)  the  universal  affirmative,  All  S  is  F,  usually  denoted 
by  the  symbol  A  ; 

(2)  the  particular  affirmative.  Some  S  is  F,  usually  de- 
noted by  the  symbol  I ; 

(3)  the  universal  fiegative,  No  S  is  F^  usually  denoted 
by  the  symbol  E  ; 

(4)  the  particular  negative.  Some  ^  is  not  F,  usually 
denoted  by  the  symbol  O. 

These  symbols  A,  I  and  E,  O  are  taken  from  the  Latin 
words  affirmo  and  nego,  the  affirmative  symbols  being  the 
first  two  vowels  of  the  former,  and  the  negative  symbols 
the  two  vowels  of  the  latter. 

Besides  these  symbols,  it  will  also  be  found  convenient 
sometimes  to  use  the  following, — 

SaF=K\\S  isF; 
SiF^  Some  S  is  F-, 
SeF=NoS  isF; 
SoF  =  Some  S  is  not  F. 

The  above  are  useful  when  we  wish  that  the  symbol 
which  is  used  to  denote  the  proposition  as  a  whole  should 
also  indicate  what  symbols  have  been  chosen  for  the  subject 
and  the  predicate  respectively.     Thus, 

MaF=  AW  M  \sF; 

FoQ  ~  Some  F  is  not  Q. 

The  universal  negative  should  be  written  in  the  form 
No  S  is  F,  not  All  S  is  not  F ;  for  the  latter  would  usually 


40 


PROPOSITIONS. 


[part  II. 


be  understood  to  be  merely  particular.  Thus,  All  that 
glitters  is  not  gold  is  really  an  O  proposition,  and  is  equi- 
valent to — Some  things  that  glitter  [  are  not  |  gold. 

39.     Indefinite  Propositions. 

According  to  Quantity,  Propositions  have  sometimes 
been  divided  into  (i)  Universal,  (2)  Particular,  (3)  Singular, 
(4)  Indefinite.  Singular  propositions  are  discussed  in  the 
following  section. 

By  an  Indefmiie  Proposition  is  meant  one  "  in  which  the 
Quantity  is  not  explicitly  declared  by  one  of  the  designatory 
terms  «//,  every,  some,  }nany,  &c."  We  may  perhaps  say 
with  Hamilton  that  indesignate  or  preiiidesignate  would  be  a 
better  term  to  employ.  There  can  be  no  doubt  that,  as 
Mansel  remarks,  "  The  true  indefinite  proposition  is  in  fact 
the  particular;  the  statement  *some  Ais  B^  being  applicable 
to  an  uncertain  number  of  instances,  from  the  whole  class 
down  to  any  portion  of  it.  For  this  reason  particular  pro- 
positions were  called  indefinite  by  Thcophrastus  "  (A/dn'c/iy 
p.  49). 

Some  indesignate  propositions  are  no  doubt  intended  to 
be  understood  as  universals,  e.g..  Comets  are  subject  to  the 
law  of  gravitation ;  but  in  such  cases  before  we  deal  with 
the  proposition  logically  it  is  better  that  the  word  a//  should 
be  explicitly  prefixed  to  it.  If  wc  are  really  in  doubt  with 
regard  to  the  quantity  of  the  proposition  it  must  logically 
be  regarded  as  particular. 

Other  designations  of  quantity  besides  a//  and  some,  e.g., 
most,  are  discussed  in  section  41. 

The  term  indefinite  has  also  been  applied  to  propositions 
in  another  sense.  According  to  Quality,  instead  of  the  two- 
fold division  given  in  the  preceding  example,  a  threefold 
division   is   sometimes   adopted,   namely  into    affirmative, 


CHAP.  I.] 


PROPOSITIONS. 


41 


negative,  and  infinite  or  indefinite.     For  further  explanation, 
see  section  44. 

40.     Singular  Propositions. 

By  a  Singular  or  Individual  Proposition  is  meant  a  pro- 
position of  w^hich  the  subject  is  a  singular  term,  one  there- 
fore in  which  the  affirmation  or  denial  is  made  but  of  a  single 
specified  individual;  e.g.,  Brutus  is  an  honourable  man; 
Much  Ado  about  Nothing  is  a  play  of  Shakespeare's;  My 
boat  is  on  the  shore. 

Singular  propositions  may  usually  be  regarded  as  forming 
a  sub-class  of  Universal  propositions,  since  in  every  singular 
proposition  the  affirmation  or  denial  is  of  the  7uhole  of  the 
subject.  Such  propositions  have  however  certain  pecu- 
liarities of  their  own,  as  we  shall  note  subsequently;  e.g.,  they 
have  not  like  other  universal  propositions  a  contrary  distinct 
from  their  contradictory. 

Hamilton  distinguishes  between  Universal  and  Singular 
Propositions,  the  predication  being  in  the  former  case  of  a 
IV/iole  Undivided,  and  in  the  latter  case  of  a  Unit  Indivisible. 
This  separation  is  sometimes  useful ;  but  I  think  it  better 
not  to  make  it  absolute.  A  singular  proposition  may  without 
risk  of  confusion  be  denoted  by  one  of  the  symbols  A  or  E  ; 
and  in  syllogistic  inferences,  a  singular  may  always  be  re- 
garded as  equivalent  to  a  universal  proposition.  The  use 
of  independent  symbols  for  affirmative  and  negative  singular 
propositions  would  introduce  considerable  additional  com- 
plexity into  the  treatment  of  the  Syllogism ;  and  for  this 
reason  alone  it  seems  desirable  as  a  rule  to  include  par- 
ticulars under  universals.  We  may  however  divide  universal 
propositions  into  General  and  Singular,  and  we  shall  then 
have  terms  whereby  to  call  attention  to  the  distinction 
wherever  it  may  be  necessary  or  useful  to  do  so. 


42 


PROPOSITIONS. 


[part  II. 


There  is  a  certain  class  of  propositions  with  regard  to 
which  there  is  some  difference  of  opinion  as  to  whether  they 
should  be  regarded  as  singular  or  particular ;  for  example, 
such  as  the  following :  A  certain  man  had  two  sons ;  A  great 
statesman  was  present.  Mansel  {Aidnchy  p.  49)  decides 
that  they  should  be  dealt  with  as  particulars,  and  I  think 
rightly,  on  the  ground  that  if  we  have  two  such  propositions, 
"  a  certain  man  "  or  "  a  great  statesman  "  being  the  subject 
of  each,  we  cannot  be  sure  that  the  same  individual  is 
referred  to  in  both  cases.  Sometimes  however  the  context 
may  enable  us  to  decide  the  case  differently. 

There  are  propositions  of  another  kind  with  a  singular 
term  for  subject  about  which  a  few  words  may  be  said ; 
namely,  such  propositions  as — Browning  is  sometimes  ob- 
scure ;  That  boy  is  sometimes  first  in  his  class.  These 
propositions  may  be  treated  as  universal  with  a  somewhat 
complex  predicate,  (and  it  should  be  noted  that  in  bringing 
propositions  into  logical  form  we  are  frequently  compelled 
to  use  very  complex  predicates) ;  thus  : — 

Browning  |  is  |  a  poet  who  is  sometimes  obscure. 

That  boy  |  is  |  a  boy  who  is  sometimes  first  in  his  class. 

By  a  certain  transformation  however  these  propositions 
may  also  be  dealt  with  as  particulars,  and  such  transforma- 
tion may  sometimes  be  convenient;  thus.  Some  of  Browning's 
writings  are  obscure,  Some  of  the  boy's  places  in  his  class 
are  the  first  places.  But  when  the  proposition  is  thus  modi- 
fied, the  subject  is  no  longer  a  singular  term. 

41.  The  logical  signification  of  the  words  sojne^ 
viost,  few,  all,  any. 

Some  may  mean  merely  "some  at  least," />.,  not  none,  or 
it  may  carry  the  further  implication,  "some  at  most,"  />.,  not 
all.     Professor   Bain   is   probably   right  in   saying   {Lo^ie, 


CHAP.  I.] 


PROPOSITIONS. 


43 


Deduction,  p.  81)  that  in  ordinary  speech  the  latter  meaning 
is  the  more  usual.  With  most  modern  logicians,  however, 
the  logical  implication  of  some  is  limited  to  some  at  least, 
not  exclusive  of  all.  Using  the  word  in  this  sense,  if  we 
want  to  express  "  some,  but  not  all,  S  is  P,"  we  must  make 
use  of  two  propositions, 

Some  S  is  P, 
Some  S  is  not  P. 

The  particular  then  is  not  exclusive  of  the  universal.  As 
already  suggested,  it  is  indefinite,  though  with  a  certain 
limit ;  that  is,  it  is  indefinite  so  far  that  it  may  apply  to  any 
number  from  a  single  one  up  to  all,  but  on  the  other  hand 
it  is  definite  so  far  as  it  excludes  "  none." 

It  may  be  added  that  in  regarding  "some"  as  implying 
no  more  than  at  least  one,  we  are  probably  again  departing 
from  the  ordinary  usage  of  language,  which  would  regard  it 
as  impl)'ing  at  least  two. 

[It  should  perhaps  be  noted  that  on  rare  occasions 
"some"  may  have  a  slightly  different  implication.  For 
example,  the  proposition  "  Some  truth  is  better  kept  to 
oneself"  may  be  so  emphasized  as  to  make  it  perfectly 
clear  to  what  particular  kind  of  truth  reference  is  made. 
This  is  however  extra-logical.  Logically  the  proposition 
must  be  treated  as  particular,  or  it  must  be  written  in 
another  form,  "All  truth  of  a  certain  specified  kind  is 
better  kept  to  oneself."  Thus,  Spalding  remarks  {Logic, 
p.  6^),  "The  logical  *some'  is  totally  indeterminate  in  its 
reference  to  the  constitutive  objects.  It  is  always  aliqui, 
never  quidam;  it  designates  some  objects  or  other  of  the 
class,  not  some  certain  objects  definitely  pointed  out."] 

Most  is  to  be  interpreted  "at  least  one  more  than  half." 
Fetu  has  a  negative  force,  "Few  S  is  /*"  being  equivalent 


44 


PROPOSITIONS. 


[part  II. 


to  "Most  S  is  not  /"';  (with  perhaps  the  further  implication 
"  although  some  S  is  -P"  ;  thus  Few  S  is  P  is  given  by  Kant 
as  an  example  of  the  exponible  proposition,  on  the  ground 
that  it  contains  both  an  affirmation  and  a  negation,  though 
one  of  them  in  a  concealed  way).  Formal  logicians  (except- 
ing De  Morgan  and  Hamilton)  have  not  as  a  rule  recognized 
these  additional  signs  of  quantity;  and  it  is  true  that  in 
many  logical  combinations  we  are  unable  to  regard  them  as 
more  than  particular  propositions,  Most  S  is  P  being  re- 
duced to  Some  S  is  /*,  and  Few  S  is  P  to  Some  Sis  not  P. 
Sometimes  however  we  are  able  to  make  use  of  the  extra 
knowledge  given  us;  e.g.,  from  Most  M  is  P,  Most  M  is 
S  we  can  infer  Some  S  is  P,  although  from  Some  Af  is  7^, 
Some  M  is  S  we  can  infer  nothing. 

It  should  be  observed  that  A  fciv  has  not  the  same 
signification  as  Fca'^  but  must  be  regarded  as  affirmative, 
and,  generally,  as  simply  equivalent  to  some;  e.g.,  A  few  S 
is  P  -  Some  S  is  P.  Sometimes,  however,  it  means  "  a  small 
number,"  and  in  this  case  the  proposition  is  perhaps  best 
regarded  as  singular,  the  subject  being  collective.  Thus 
*'  a  few  peasants  successfully  defended  the  citadel "  may  be 
rendered  "  a  small  band  of  peasants  successfully  defended 
the  citadel,"  rather  than  "some  peasants  successfully  defended 
the  citadel,"  since  the  stress  is  intended  to  be  laid  at  least 
as  much  on  the  paucity  of  their  numbers  as  on  the  fact 
that  they  were  peasants.  In  this  case,  the  proposition  would 
be  A,  not  I. 

It  may  here  be  remarked  that  in  all  cases,  where  we  are 
dealing  with  propositions  which  as  originally  stated  are  not 
in  a  logical  form,  the  first  problem  in  reducing  them  to 
logical  form  is  one  of  interpretation,  and  we  must  not  be 
surprised  to  find  that  in  many  cases  different  methods  of 
interpretation  lead  to  different  results.     No  confusion  will 


CHAP.  I.] 


PROPOSITIONS. 


45 


ensue  if  we  make  it  perfectly  clear  what  we  do  regard  as 
the  logical  form  of  the  proposition,  and  also  how  we  have 
arrived  at  our  result. 

AUis  ambiguous,  so  far  as  it  may  be  used  either  dis- 
tributively  or  collectively.  In  the  proposition  "All  the 
angles  of  a  triangle  are  less  than  two  right  angles "  it  is 
used  distributively,  the  predicate  applying  to  each  and 
every  angle  of  a  triangle  taken  separately.  In  the  propo- 
sition "  All  the  angles  of  a  triangle  are  equal  to  two  right 
angles  "  it  is  used  collectively,  the  predicate  applying  to  all 
the  angles  taken  together,  and  not  to  each  separately. 

A7iy  as  the  sign  of  quantity  of  the  subject  of  a  cate- 
gorical proposition,  (e.g.,  any  S  is  P),  is  logically  equivalent 
to  "all"  in  its  distributive  sense.  Whatever  is  true  of  any 
member  of  a  class  taken  at  random  is  necessarily  true  of 
the  whole  of  that  class.  When  not  the  subject  of  a  cate- 
gorical proposition,  any  may  have  a  different  signification. 
For  example,  in  the  hypothetical  proposition,  If  any  A  is  B, 
C  is  Z>,  it  has  the  same  indefinite  character  which  we 
logically  ascribe  to  "some";  since  the  antecedent  condition 
is  satisfied  if  a  single  A  is  B.  The  proposition  might  indeed 
be  written — If  one  or  more  A  is  B,  C  is  D. 

42.     Examine  the  logical  signification  of  the  itali- 
cised words  in  the  following  propositions : — 

SojHc  arc  born  great. 

Fezu  are  chosen. 

All  is  not  lost. 

All  men  are  created  equal. 

-^//that  a  man  hath  will  he  give  for  his  life. 

1(  some  A  is  B,  some  Cis  D. 

If  any  A  is  B,  any  C  is  D, 

Hall  AisB,  all  CisD. 


46 


PROPOSITIONS. 


[part  II. 


43.  Distinguish  the  collective  and  distributive 
use  of  the  word  all  in  the  following  propositions  : 

(i)  AH  Albinos  are  pink-eyed  people  ; 

(2)  Omncs  apostoli  sunt  duodecim ; 

(3)  Non  omnis  moriar  ; 

(4)  Non  omnia  possumus  omnes  ; 

(5)  All  men  find  their  own  in  all  men's  good, 
And  all  men  join  in  noble  brotherhood. 

(6)  Not  all  the  gallant  efforts  of  the  officers  and 
escort  of  the  British  Embassy  at  Cabul  were  able  to 
save  them. 

[Jevons,  Elementary  Lessons  in  Logic,  p.  297.  Studies 
in  Deductive  Logic,  pp.  19,  28.] 

44.  Lnjinitc  or  indefinite  terms  and  propositions. 

Infinite  and  indefinite  are  designations  applied  to  terms 
having  a  thoroughgoing  negative  character ;  to  such  a  term 
for  example  as  "not-white,"  understood  as  denoting  not 
merely  coloured  things  other  than  white,  but  the  whole 
infinite  or  indefinite  class  of  things  of  which  "  white  "  can- 
not be  truly  affirmed,  including  such  entities  as  Mill's  Logic, 
a  dream,  Time,  a  soliloquy.  New  Guinea,  the  Seven  Ages 
of  Man. 

It  is  however  to  be  observed  that  if  symbols  are  used, 
it  is  impossible  to  say  which  of  the  terms  S  or  not-5  really 
partakes  of  this  indefinite  character,  since,  for  example, 
there  is  nothing  to  prevent  our  having  originally  written  S 
for  "not-white,"  in  which  case  "white"  becomes  not-5, 
and  S  is  the  really  indefinite  or  infinite  term. 

Following  out  the  above  idea,  propositions  were  divided 
by  Kant  into  three  classes  in  respect  of  Quality,  namely, 
affirmative — A  is  B,  negative — A  is  not  B,  and  infinite  (or 


CHAP.  I.] 


PROPOSITIONS. 


47 


indefinite)— A  is  not-^.  Logically  however  the  last  proposi- 
tion (which  is  equivalent  to  the  second  in  meaning)  must  be 
regarded  as  simply  affirmative.  Asjust  shewn,  it  is  impossible 
to  say  which  of  the  terms  B  or  not-^  is  really  infinite  or  in- 
definite ;  and  it  is  therefore  also  impossible  to  say  which  of 
the  propositions  "^  is  B''  or  "^  is  not-^"  is  really  infinite 
or  indefinite.  Logically  then  they  must  be  regarded  as  be- 
longing to  the  same  type  of  proposition,  and  we  have  to 
fall  back  upon  the  two-fold  division  into  afldrmative  and 
negative^ 

45.     Can  distinctions  of  Quality  and  Quantity  be 
applied  to  Hypothetical  and  Disjunctive  Propositions.? 

The  parts  of  the  Hypothetical  Proposition  are  called  the 
Antecedent  and  the  Consequent.  Thus,  in  the  proposition, 
"  If  A  is  B,  C  is  Z>,"  the  Antecedent  is  ''A  is  B/'  the 
Consequent  is  "  C  is  L>".  The  Quality  of  the  Hypothetical 
Proposition  depends  upon  the  Quality  of  the  Consequent. 
Thus,  the  proposition  If  A  is  B,  C  is  not  Z>,  is  to  be 
considered  negative.  Hypothetical  propositions  may  also 
be  regarded  as  Universal  or  Particular,  according  as  the 
consequent  is  afllirmed  to  follow  from  the  antecedent  in  all 
or  only  in  some  cases.  We  have  then  the  four  fundamental 
types  of  proposition  : — 

(i)  If  A  is  B,  CisD.  A. 

(2)  In  some  cases  in  which  A  is  B,  C  is  D,  I. 

(3)  If^  is  j5,  CisnoiD.  e. 

(4)  In  some  cases  in  which  A  is  B,  C  is  not  L>,  O. 

^  It  should  be  observed  that,  if  we  admit  its  use  as  above,  the  term 
indrfinite  as  applied  to  propositions  is  ambiguous,  since  by  an  indefinite 
proposition  we  mean  here  something  entirely  different  from  what  was 
called  an  indefmite  proposition  in  section  39.  In  the  one  case  the 
reference  is  to  the  Quality  of  the  proposition,  in  the  other  case  to  its 
(Quantity. 


48 


PROPOSITIONS. 


[part  II. 


The  student  must  be  warned  against  treating  such  a 
proposition  as  *' If  any  A  is  B,  some  C  is  Z>"  as  par- 
ticular'. Regarded  separately  the  antecedent  and  the  con- 
sequent in  this  example  are  both  particular;  but  the  con- 
nection between  them  is  affirmed  universally,  the  proposition 
asserting  that  "/>/  all  cases  in  which  any  A  is  B,  some  C 

is  nr 

It  should  be  observed  that  in  a  considerable  number  of 
cases,  the  hypothetical  is  of  the  nature  of  a  singular  pro- 
position, the  event  referred  to  in  the  antecedent  being  in 
the  nature  of  things  one  which  can  happen  but  once;  e.g., 
If  I  perish  in  the  attempt,  I  shall  not  die  unavenged. 

To  the  Disjunctive  Proposition  we  are  unable  to  apply 
distinctions  of  Quality.  The  proposition,  Neither  F  is  Q 
nor  X  is  Y  states  no  alternative,  and  is  therefore  not  dis- 
junctive at  all.  Distinctions  of  Quantity  are  however  still 
applicable.     Thus, 

Universal, — Either  Pi?,  ^  or  X is  K 

Particular, — In  some  cases  either  Pis  Qor  X is  Y. 

It  is  again  to  be  observed  that  frequently  the  dis- 
junctive proposition  is  of  the  nature  of  a  singular  proposi- 
tion, the  reference  being  but  to  a  single  occasion  on  which 
it  is  asserted  that  one  of  the  alternatives  will  hold  good. 

46.  Determine  the  Quantity  and  Quality  of  the 
following  propositions,  stating  precisely  what  you 
regard  as  the  subject  and  predicate,  or  in  the  case 


^  I  cannot  ngrcc  with  Hamilton  [Logic,  i.  p.  248),  in  regarding  the 
following  as  a  particular  hypothetical — If  some  Dodo  is,  then  some 
animal  is.  The  proposition  is  a  little  hard  to  interpret,  but  it  seems  to 
mean  that  if  there  is  such  a  thing  as  a  Dodo,  then  there  is  such  a  thing 
as  an  animal ;  and  we  must  consider  that  a  universal  connection  is  here 
affirmed. 


CHAP.  I.]  PROPOSITIONS.  4^ 

of   hypothetical    propositions,    the    antecedent    and 
consequent  of  each  : — 

(i)     All  men  think  all   men   mortal  but   them- 
selves. 

(2)  Not  to  know  mc  argues  thyself  unknown. 

(3)  To  bear  is  to  conquer  our  fate. 

(4)  Berkeley,   a   great   philosopher,    denied   the 
existence  of  Matter. 

(5)  A  great  philosopher  has  denied  the  existence 
of  Matter. 

(6)  The  virtuous  alone  arc  happy. 

(7)  None  but  Irish  were  in  the  artillery. 

(8)  Not  every  talc  we  hear  is  to  be  believed. 

(9)  Great  is  Diana  of  the  P:phesians  ! 

(10)  All  sentences  arc  not  propositions. 

(11)  Where  there's  a  w^ill  there's  a  way. 

(12)  Some  men  are  always  in  the  wrong. 

(13)  Facts  arc  stubborn  things. 

(14)  He    that    incrcaseth    knowledge   increascth 
sorrow. 

(15)  None  think  the  great  unhappy,  but  the  great. 

(16)  He  can't  be  wrong,  whose  life  is  in  the  right. 

(17)  Nothing  is  expedient  which  is  unjust. 

(18)  Mercy   but    murders,   pardonin^r  those   that 
kill. 

(19)  If  virtue  is  involuntary,  so  is  vice. 

(20)  Who  spareth  the  rod,  hateth  his  child. 

47.  Analyse  the  following  propositions,  i.e.,  ex- 
press them  in  one  or  more  of  the  strict  categorical 
forms  admitted  in  Logic  : — 

K.  L. 


50  PROPOSITIONS.  [part  ii. 

(i)  No  one  can  be  rich  and  happy  unless  he  is 
also  temperate  and  prudent,  and  not  always  then. 

(ii)  No  child  ever  fails  to  be  troublesome  if  ill 
taught  and  spoilt. 

(iii)  It  would  be  equally  false  to  assert  that  the 
rich  alone  are  happy,  or  that  they  alone  are  not.  [v.] 

(i)  contains  two  statements  which  may  be  reduced  to 
the  following  forms, — 

All  who  arc  rich  and  happy  |  are  [  temperate  and 
prudent.     A. 

Some  who  are  temperate  and  prudent  ]  arc  not  |  rich 
and  happy.     O. 

(ii)  may  be  written,  All  ill-taught  and  spoilt  children 
are  troublesome.     A. 

(iii)  Here  two  statements  are  given  false,  namely,  the 
rich  alone  are  happy ;  the  rich  alone  are  not  happy. 

We  may  reduce  these  false  statements  to  the  following, — 
all  who  are  happy  are  rich  ;  all  who  are  not  happy  are  rich. 
And  this  gives  us  these  true  statements, — 

Some  who  are  happy  are  not  rich.     O. 
Some  who  are  not  hai)i)y  are  not  rich.     O. 
The  original  proposition  is  expressed  tlierefore  by  means 
of  these  two  particular  negative  propositions. 


48.     The  Distribution  of  Terms  in  a  Proposition. 

A  term  is  said  to  be  distributed  when  reference  is  made 
to  all  the  individuals  denoted  by  it ;  it  is  said  to  be  undis- 
tributed when  they  are  only  referred  to  partially,  i.e.,  in- 
formation is  given  with  regard  to  a  portion  of  the  class 
denoted  by  the  term,  but  we  are  left  in  ignorance  with 
regard  to  the  remainder  of  the  class.    It  follows  immediately 


CHAP.  I.J 


PROPOSITIONS. 


51 


from  this  definition  that  the  subject  is  distributed  in  a 
universal,  and  undistributed  in  a  particular,  proposition. 
It  can  further  be  shewn  that  the  predicate  is  distributed  in 
a  negative,  and  undistributed  in  an  affirmative  proposition. 
Thus,  if  I  say,  All  .S  is  Z',  I  imply  that  at  any  rate  some  P  is 
S,  but  I  make  no  implication  with  regard  to  the  whole  of  P. 
I  leave  it  an  open  question  as  to  whether  there  is  or  is  not 
any  P  outside  the  class  S.  Similarly  if  I  say,  Some  S  is  P. 
But  if  I  say,  No  S  is  P,  in  excluding  the  whole  of  S  from  P, 
I  am  also  excluding  the  whole  of  P  from  S,  and  therefore  P 
as  well  as  S  is  distributed.  Again,  if  I  say.  Some  6"  is  not  /*, 
although  I  make  an  assertion  with  regard  to  a  part  only  of 
S,  I  exclude  this  part  from  the  w'hole  of  P,  and  therefore 
the  w^hole  of  P  from  it.  In  this  case,  then,  the  predicate  is 
distributed,  although  the  subject  is  not. 

Summing  up  our  results  we  find  that 

A  distributes  its  subject  only, 

I  distributes  neither  its  subject  nor  its  predicate, 

E  distributes  both  its  subject  and  its  predicate, 

O  distributes  its  predicate  only. 

49.  How  docs  the  Quality  of  a  Proposition  affect 
its  Quantity  ?     Is  the  relation  a  necessary  one  }     [L.] 

By  the  Quantity  of  a  Proposition  must  here  be  meant 
the  Quantity  of  its  Predicate,  and  we  have  shewn  in  the 
preceding  section  that  this  is  determined  by  its  Quality. 
The  predicate  is  distributed  in  negative,  undistributed  in 
affirmative,  propositions. 

The  latter  part  of  the  above  question  refers  to  Hamilton's 
doctrine  of  the  Quantification  of  the  Predicate.  According 
to  this  doctrine,  the  predicate  of  an  affirmative  proposition 
is  sometimes  expressly  distributed,  while  the  predicate  of  a 

4—2 


52 


PROPOSITIONS. 


[part  II. 


negative  proposition  is  sometimes  given  undistributed.     For 
example,  the  following  forms  are  introduced  : — 

Some  S  is  all  P, 
No  5  is  some  P. 

This    doctrine    is    discussed   and    illustrated   in    Part    in. 
chapter  9. 

50.  In  doubtful  cases  how  should  you  decide 
which  is  the  subject  and  which  the  predicate  of  a 
proposition  ?  [v.] 

The  nature  of  the  distinction  between  the  subject  and 
the  predicate  of  a  i)ro[)osition  may  be  expressed  by  saying 
that  the  subject  is  that  of  which  something  is  affirmed  or 
denied,  the  predicate  is  that  which  is  affirmed  or  denied  of 
the  subject;  or  perhaps  still  better,  the  subject  is  that  which 
we  think  of  as  the  determined  or  qualified  notion,  the 
predicate  that  which  we  think  of  as  the  determining  or 
qualifying  notion. 

Now,  can  we  say  that  the  subject  always  precedes  the 
copula,  and  that  the  predicate  always  follows  it?  In 
other  words,  can  we  consider  the  order  of  the  terms  to 
suffice  as  a  criterion?  If  the  proposition  is  reduced  to 
an  equation,  as  in  the  doctrine  of  the  quantification  of  the 
predicate,  I  do  not  see  what  other  criterion  we  can  take; 
or  we  might  rather  say  that  in  this  case  the  distinction 
between  subject  and  predicate  itself  fails  to  hold  good. 
The  two  are  placed  on  an  equality,  and  we  have  nothing 
left  by  which  to  distinguish  them  except  the  order  in  which 
they  are  stated.  This  view  is  indicated  by  Professor  Baynes 
in  his  Essay  on  the  Nau  Ajtalyiic  of  Logical  Forms.  In  such 
a   proposition,  for  example,  as   "Great   is   Diana   of  the 


CHAP.  I.] 


PROPOSITIONS. 


53 


Ephesians,"  he  would  call  "great"  the  subject,  reading  the 
proposition,  however,  "  (Some)  great  is  (all)  Diana  of  the 
Ephesians." 

But  leaving  this  view  on  one  side,  we  cannot  say  that 
the  order  of  terms  is  always  a  sufficient  criterion.  In  the 
proposition  just  quoted,  "Diana  of  the  Ephesians "  would 
generally  be  accepted  as  the  subject.  What  further  criterion 
then  can  be  given?  In  the  case  of  E  and  I  propositions, 
(propositions,  as  will  be  shewn,  which  can  be  simply  con- 
verted), we  must  appeal  to  the  context  or  to  the  question  to 
which  the  proposition  is  an  answer.  If  one  term  clearly 
conveys  information  regarding  the  other  term,  it  is  the 
predicate.  It  is  also  more  usual  that  the  subject  should  be 
read  in  extension  and  the  predicate  in  intension.  If  none 
of  these  considerations  are  decisive,  then  I  should  admit 
that  the  order  of  the  terms  must  suffice.  In  the  case  of 
A  and  O  propositions,  (propositions,  as  will  be  shewn, 
which  cannot  be  simply  converted),  a  further  criterion  may 
be  added.  From  the  rules  relating  to  the  distribution  of 
terms  in  a  proposition  it  follows  that  in  affirmative  pro- 
positions the  distributed  term,  (if  either  term  is  distributed), 
is  the  subject ;  whilst  in  negative  propositions,  if  only  one 
term  is  distributed,  it  is  the  predicate.  I  am  not  sure  that 
the  inversion  of  terms  ever  occurs  in  the  case  of  an  O  pro- 
position ;  but  in  A  propositions  it  is  not  infrequent.  Ap- 
plying the  above  to  such  a  proposition  as  "  Workers  of 
miracles  were  the  apostles,"  it  is  clear  that  the  latter  term 
is  distributed  while  the  former  is  not.  The  latter  term  is 
therefore  the  subject.  A  corollary  from  the  rule  is  that  in 
an  affirmative  proposition  if  one  and  only  one  term  is 
singular  that  is  the  subject,  since  a  singular  is  ecjuivalent 
to  a  distributed  term.  This  decides  such  a  case  as  "  Great 
is  Diana  of  the  Ephesians.'' 


54  PROPOSITIONS.  [PART  il. 

51.  What  do  you  consider  to  be  respectively  the 
subject  and  the  predicate  of  the  following  sentences, 
and  why  ? 

(i)  Few  men  attain  celebrity. 

(2)  Blessed  are  the  peacemakers. 

(3)  It  is  mostly  the  boastful  who  fail. 

(4)  Clematis  is  Traveller's  Joy.  [v.] 

52.  What  do  you  consider  to  be  the  essential 
distinction  between  the  Subject  and  Predicate  of  a 
proposition  ?    Apply  your  answer  to  the  following  : — 

(i)     From  thence  thy  warrant  is  thy  sword. 
(2)     That  is  exactly  what  I  wanted.  [v.] 


CHAPTER  II. 

THE   OPPOSITION   OF   PROPOSITIONS. 

53.     The  Opposition  of  Categorical  Propositions. 

Two  propositions  are  said  to  be  opposed  to  each  other 
when  they  have  the  same  subject  and  predicate  respectively, 
but  differ  in  quantity  or  quality  or  both  ^ 

Taking  the  propositions  SaP^  SiP,  SePy  SoP,  in  pairs 
we  find  that  there  are  four  j^ossible  kinds  of  relation  between 
them. 

(i)  The  pair  of  propositions  may  be  such  that  they 
cannot  both  be  true,  and  they  cannot  both  be  false.  This 
is  called  contradictory  opposition,  and  subsists  between 
SaP  and  SoP^  and  between  SeP  and  SiP. 


^  This  definition  is  given  by  Aldrich  (p.  53  in  Mansel's  edition). 
Ueberweg  however  defines  Opposition  in  such  a  way  as  to  include 
only  contradiction  and  contrariety  (translation  by  Lindsay,  p.  328); 
and  Mansel  remarks  that  "Subalterns  are  improperly  classed  as  opposed 
propositions"  {Aldrich^  p.  59).  Professor  Fowler  follows  Aldrich's 
definition  {Deductive  Logic^  p.  74),  and  I  think  wisely.  We  want  some 
term  to  signify  this  general  relation  between  propositions ;  and  though 
it  might  be  possible  to  find  a  more  convenient  term,  I  do  not  think 
that  any  confusion  is  likely  to  result  from  the  use  of  the  term  opposition 
if  the  student  is  careful  to  notice  that  it  is  here  used  in  a  technical 
sense. 


56 


PROPOSITIONS. 


[part  II. 


(2)  They  may  be  such  that  they  cannot  both  be  true, 
but  they  may  both  be  false.  This  is  called  contrary  oppo- 
sition.    SaF  and  SeP. 

(3)  They  may  be  such  that  they  cannot  both  be  false, 
but  they  may  both  be  true.  SubcorJrary  opposition.  SiP 
and  SoP. 

(4)  From  a  given  universal  proposition,  the  truth  of 
the  particular  having  the  same  quality  follows,  but  not  vice 
versa.  This  is  stibaliern  opposition^  the  universal  being 
called  the  sitbaltematit^  and  the  particular  the  siihaltcrjiate 
or  the  subaltern.     SaP  and  SiP.     SeP  and  SoP. 

All  these  relations  are  indicated  clearly  in  the  ancient 
square  of  opposition. 


Contraries 


.o'h 


in 
cr 

r-t- 

s 


V- 


c° 


,-c^^ 


o. 


y-> 


/a 


c 

t-l 
CO 


Siibcontraries 


O 


Propositions  must  of  course  be  brought  to  such  a  form 
that  they  have  the  same  subject  and  the  same  predicate 
before  we  can  apply  the  terms  of  opposition  to  them ;  for 
example,  All  6"  is  P  and  Some  P  is  not  S  arc  not  contra- 
dictories. 


CHAP.  II.] 


PROPOSITIONS. 


57 


54.  On  the  common  view  of  the  opposition  of 
propositions  what  are  the  inferences  to  be  drawn 
(i)  from  the  truth,  (2)  from  the  falsity,  of  each  of  the 
four  categorical  propositions  ?  [l.] 

55.  Explain  the  nature  of  the  opposition  between 
each  pair  of  the  following  propositions  : 

None  but  Liberals  voted  against  the  motion. 

Amongst  those  who  voted  against  the  motion  were 
some  Liberals. 

It  is  untrue  that  those  who  voted  against  the 
motion  were  all  Liberals. 

56.  Give  the  contradictory  and  the  contrary  of 
the  following  propositions:  — 

(i)  A  stitch  in  time  saves  nine. 

(2)  None  but  the  brave  deserve  the  fair. 

(3)  He  can't  be  wrong  whose  life  is  in  the  right. 

(4)  The  virtuous  alone  are  happy. 

(i)  A  stitch  in  time  saves  nine.  This  is  to  be  regarded 
as  a  universal  affirmative  proposition,  and  we  therefore  have 

Contradictory,  Some  stitches  in  time  do  not  save  nine.    I. 

Contrary,  No  stitch  in  time  saves  nine.     E. 

(2)  None  but  the  brave  deserve  the  fair,  =  None  who 
are  not  brave  deserve  the  fair.     E. 

Contradictory,  Some  who  are  not  brave  deserve  the 
fair.     I. 

Contrary,  All  who  are  not  brave  deserve  the  fair.     A. 


58  PROPOSITIONS.  [part  ii. 

(3)  He  can't  be  wrong  whose  life  is  in  the  right.     E. 

Contradictory^  Some  may  be  wrong  whose  lives  are  in 
the  right.     I. 

Contrary,  All  are  wrong  whose  Hvcs  are  in  the  right.    A. 

(4)  The  virtuous  alone  are  Happy,  =  No  one  who  is  not 
virtuous  is  happy.     E. 

Contradictory,  Some  who  are  not  virtuous  are  happy.    I. 

Contrary,  All  who  are  not  virtuous  are  happy.     A. 

57.  Give  the  contrary,  contradictory,  and  sub- 
altern of  the  following  propositions  : — 

(i)  All  B.A.'s  of  the  University  of  London  have 
passed  three  examinations. 

(2)  All  men  are  sometimes  thoughtless. 

(3)  Uneasy  lies  the  head  that  wears  a  crown. 

(4)  The  whole  is  greater  than  any  of  its  parts. 

(5)  None  but  solid  bodies  are  crystals. 

(6)  He  who  has  been  bitten  by  a  serpent  is 
afraid  of  a  rope. 

(7)  lie  who  tries  to  say  that  which  has  never 
been  said  before  him  will  probably  say  that  which 
will  never  be  repeated  after  him. 

[Jevons,  Stiuiics  in  Deductive  Logic,  p.  58.] 

58.  Explain  the  technical  terms  "  contrary  "  and 
"  contradictory,"  appl)'ing  them  to  the  following  pro- 
positions : — 

(i)     Few  S  are  P. 

(2)  At  any  rate,  he  was  not  the  only  one  who 
cheated. 

(3)  Two-thirds  of  the  army  arc  abroad.        [v.] 


CHAP.  II.] 


PROPOSITIONS. 


59 


It  is  the  same  thing  to  deny  the  truth  of  a  proposition 
and  to  affirm  the  truth  of  its  contradictory ;  and  vice  versa. 
The  criterion  of  contradictory  opposition  is  that  of  the  tivo 
propositions,  one  must  be  true  and  the  other  must  be  false ; 
they  cannot  be  true  together,  but  on  the  other  hand  no 
mean  is  possible  between  them.  The  relation  between  two 
contradictories  is  mutual ;  it  docs  not  matter  which  is  given 
true  or  false,  we  know  that  the  other  is  false  or  true  ac- 
cordingly. Every  proposition  has  its  .ontradictory,  which 
may  however  be  more  or  less  complex  iU  form. 

The  contrary  of  a  given  proposition  goes  beyond  mere 
denial,  and  sets  up  a  further  assertion  as  far  as  possible 
removed  from  the  original  assertion.  It  declares  not  merely 
the  flilsity  of  the  original  proposition  taken  as  a  whole,  but 
the  falsity  of  every  part  of  it. 

It  follows  that  if  we  cannot  go  beyond  the  simple  denial 
of  the  truth  of  a  proposition,  then  it  has  no  contrary  distinct 
from  its  contradictory.  For  example,  in  order  simply  to  deny 
the  truth  of  "some  6"  is  /V'  it  is  necessary  to  affirm  that 
"  no  S  is  P,''  and  it  is  impossible  to  go  further  than  this  in 
opposition  to  the  given  proposition.  "Some  S  is  /■*"  has 
therefore  no  contrary  as  distinguished  from  its  contradictory. 

We  may  now  apply  the  terms  in  question  to  the  given 
propositions : — 

( I )  "  Few  S  are  P"  - ''  Most  S  are  not  Z',"  and  we  might 
hastily  be  inclined  to  say  that  the  contradictory  is  "  Most  .S' 
are  /*."  Both  these  propositions  would  however  be  false  in 
the  case  in  which  exactly  one  half  S  was  P.  The  true 
contradictory  therefore  is  "At  least  one  half  6*  is  P."  The 
contrary  is  "All  S  is  P^  Similarly  the  contradictory  of 
"  Most  S  are  7"'  is  "  At  least  one  half  S  is  not  7^";  and  its 
contrary  "  No  ^  is  P." 

These  examples  shew  that  if  we  once  travel  outside  the 


6o 


PROPOSITIONS. 


[part  II. 


limits  set  by  the  old  logic,  and  recognise  the  signs  of 
quantity  most  and  few  as  well  as  all  and  some^  we  soon 
become  involved  in  numerical  statements.  Propositions  of 
the  above  kind  arc  therefore  usually  relegated  to  what  has 
been  called  numerical  logic,  a  topic  discussed  at  length  by 
De  Morgan  and  to  some  extent  by  Jevons. 

(2)  "At  any  rate,  he  was  not  the  only  one  who  cheated." 
A  question  of  interpretation  is  naturally  raised  here ;  does 
the  statement  assert  that  he  cheated,  or  is  this  left  an  open 
question  .>  We  may  I  think  choose  the  latter  alternative. 
What  the  speaker  intends  to  lay  stress  upon  is  that  some 
others  cheated  at  any  rate,  whatever  may  have  been  the 
case  with  him.  The  contradictory  then  becomes  "No  others 
cheated";  and  we  have  no  distinct  contrary. 

(3)  "Two-thirds  of  the  army  are  abroad."  This  may 
mean  "At  least  two-thirds  of  the  army  are  abroad,"  or 
"Exactly  two-thirds  of  the  army  are  abroad." 

On  the  first  interpretation,  the  contradictory  is  "Less 
than  two-thirds  of  the  army  are  abroad";  and  the  contrary 
"None  of  the  army  are  abroad." 

On  the  second  interpretation,  the  contradictory  is  "  Not 
exactly  two-thirds  of  the  army  are  abroad,"  i.e.,  "Either 
more  or  less  than  two-thirds  of  the  army  are  abroad." 
With  regard  to  the  contrary  we  are  in  a  certain  diiticulty; 
for  we  may  as  it  were  proceed  in  two  directions,  and  take 
our  choice  between  "All  the  army  are  abroad"  and  "None 
of  the  army  are  abroad."  I  hardly  see  on  what  ])rinciple 
we  are  to  choose  between  these. 

Fortunately,  contrary  opposition,  unlike  contradictory 
opposition,  is  of  very  little  logical  importance. 

59.     The  Opposition  of  Singular  Propositions. 
Take  the  proposition,  Socrates  is   wise.     The   contra- 


CHAP.  II.] 


PROPOSITIONS. 


61 


dictory  is — Socrates  is  not  wise ;  and  so  long  as  we  keep  to 
the  same  terms,  we  cannot  go  beyond  this  simple  denial. 
We  have  therefore  no  contrary  distinct  from  the  contra- 
dictory. This  opposition  of  singulars  has  been  called 
secondary  contradiction  (jVIansel's  Aldrich,  p.  56). 

There  are  indeed  two  methods  of  treatment  according  to 
which  we  might  find  a  distinct  contrary  and  contradictory  in 
the  case  of  singular  propositions,  but  I  think  that  the  above 
treatment  according  to  which  they  are  not  distinguished  is 
preferable  to  either. 

(i)  We  might  introduce  the  material  contrary  of  the  pre- 
dicate instead  of  its  mere  contradictory,  (compare  section 
2Z).     Thus  we  should  have — 

Original  proposition,  Socrates  is  wise  ; 
Contradictory,  Socrates  is  not  wise  ; 
Contrary,  Socrates  has  not  a  grain  of  sense. 

This  might  be  called  the  material  contrary  of  the  given 
proposition'.  A  fresh  term  is  introduced  that  could  not  be 
formally  obtained  out  of  the  given  proposition.  It  still 
remains  true  that  the  singular  proposition  has  no  formal 
contrary  distinct  from  its  contradictory. 

(2)  Some  principle  of  separation  into  parts  might  be 
introduced  according  to  which  the  subject  would  be  no 
longer  a  whole  indivisible  ;  for  example,  Socrates  might  be 
regarded  as  having  different  characteristics  at  different  times 
or  under  different  conditions.  The  original  proposition 
would  then  be  read  Socrates  is  always  wise,  and  the  contra- 
dictory would  be  Socrates  is  sometimes  not  wise,  while  the 
contrary  would  be  Socrates  is  never  wise.  Treated  in  this 
manner,  however,  the  proposidon  hardly  remains  a  really 
singular  proposition. 

^  The  same  distinction  might  be  applied  to  general  propositions. 


62 


PROPOSITIONS. 


[part  II. 


60.  Can  the  ordinary  doctrine  of  the  opposition 
of  propositions  be  applied  to  hypothetical  and  dis- 
junctive propositions  ? 

It  has  been  already  shewn  that  tlie  ordinary  distinctions 
of  quantity  and  quality  may  be  applied  to  Hypothetical 
Propositions,  and  it  follows  that  the  ordinary  doctrine  of 
opposition  will  also  apply  to  them.     We  have 

UA  is/y,  C:isZ>.     A. 

In  some  cases  in  which  A  h  Bj  C  is  Z>.     I. 

If^  is^,  Cisnot /?.     E. 

In  some  cases  in  which  A  is  B,  C  is  not  Z>.     O. 

Then,  as  in  the  case  of  Categoricils, — • 
A  and  I,  E  and  O  are  subalterns. 
A  and  E  are  contraries. 
A  and  O,  E  and  I  are  contradictories. 
I  and  O  are  subcontraries. 

There  is  more  danger  of  contradictories  being  confused 
with  contraries  in  the  case  of  Hypotheticals  than  there  is 
in  the  case  of  Categoricals.  /f  A  is  B^  C  is  not  D  is  very 
liable  to  be  given  as  the  contradictory  oi  If  A  /y  7?,  C  is 
D.  lUit  it  clearly  is  not  its  contradictory,  so  fr.r  as  they 
arc  general  propositions^  since  both  may  be  false.  For  ex- 
ample, the  two  statements, — If  the  Times  says  one  thing, 
the  Pall  Mall  says  another;  If  the  Times  says  one  thing, 
the  Pall  ^lall  says  the  same,  />.,  docs  not  say  another, — are 
both  false  :  the  two  papers  arc  sometimes  in  agreement  and 
sometimes  not. 

If  however  the  Hypothetical  proposition  is  of  the  nature 
of  a  Singular,  that  is,  if  the  diing  referred  to  in  the  ante- 
cedent can  happen  but  once;  then  as  in  the  case  of  Singular 
(Categorical  propositions,  the  Contradictory  and  the  Contrary 
are  not  to  be  distinguished.     Taking  the  proposition — If  I 


CHAP,  ii.l 


PROPOSITIONS. 


^^ 


perish  in  the  attempt,  I  shall  not  die  unavenged ;  its  con- 
tradictory may  fairly  be  stated— If  I  perish  in  the  attempt, 
I  shall  die  unavenged. 

We  cannot  apply  distincdons  of  quality  to  Disjunctives, 
and  therefore  the  ordinary  doctrine  of  opposition  cannot  be 
applied  to  them.  We  may  however,  find  the  contradictory 
and  the  contrary  of  a  disjunctive  proposition,  such  as  A  is 
either  B  or  C.  Its  Contradictory  is—In  some  cases  A  is 
neither  ^  nor  C;  its  Contrary—^  is  neither  B  nor  C. 
We  observe  then  that  the  contradictory  and  contrary  of  a 
disjunctive  are  not  themselves  disjunctive.  What  has  been 
said  with  regard  to  Singular  Hypotheticals  also  applies 
viutaiis  mutandis  to  what  may  be  called  Singular  Dis- 
junctives. 

A  point  to  which  our  attention  is  called  by  the  above  is 
that  the  relation  of  reciprocity  that  holds  between  contra- 
dictories does  not  always  hold  between  contraries.  If  the 
proposition  /?  is  the  contradictory  of  the  proposition  a,  then 
a  is  also  the  contradictory  of/?;  but  if  S  is  the  contrary  of  a, 
it  does  not  necessarily  follow  that  a  is  the  contrary  of  8. 
Thus,  we  have  seen  that  the  contrary  of  ''A  is  cither  B  or 
C"  is  ''A  is  neither  B  nor  C."  The  contrary  of  the  latter 
however  is  "  A  is  both  13  and  C,"  which  is  not  the  original 
proposition  over  again  \ 

61.  How  would  you  apply  the  terms  contradictory 
and  contrary  to  the  case  of  complex  propositions: 
e.g..  He  was  certainly  stupid;  and,  if  not  mad,  either 
miserably  trained,  or  misled  by  bad  companions  >  [v.] 

The  criterion  of  contradictories  given   in  section  58, 
may  be  applied  to  the  case  of  complex  propositions.     For 
example,  take  the  complex  proposition  X  is  both  A  afid  B, 
^  Cf.  also  the  Examples  given  in  section  58. 


{: 


64  PROPOSITIONS.  [part  ii. 

(where  X  is  a  singular  term).  Regarded  as  a  whole,  this 
statement  is  evidently  false  if  X  fails  to  be  either  one  or 
the  other  of  A  and  B.  It  is  also  clear  that  it  must  either 
be  both  of  them  or  it  must  fail  to  be  at  least  one  of  them. 
We  have  then  this  pair  of  contradictories, — 

\X  is  both  A  and  B  ; 

\X  is  either  not  A  or  not  B. 

Thus,  what  we  may  perhaps  call  a  conjunctive  is  contra- 
dicted by  a  disjunctive,  and  vice  veisa. 
Next  take  the  rather  more  complex  proposition — 

X  is  A,  and  cither  B  or  C\ 
Its  contradictory,  following  the  above  rule,  is 

X  is  either  not  A  or  neither  B  nor  C. 
Next  take  the  proposition 

X  is  V'j  and  if  it  is  not  Zy  it  is  cither  Q  or  B\ 
It  may  be  reduced  to 

Jf  is  V;  and  either  Z,  Q  or  /^^ 
and  we  at  once  get  the  contradictory 

X  is  either  not  Vov  neither  Z,  Q  nor  J^. 

It  will  be  noticed  that  the  last  example  chosen  is  equi- 
valent to  the  one  given  in  the  question,  the  terms  of  the 
latter  being  translated  into  symbols.  The  required  contra- 
dictory is  therefore — 

Either  he  was  not  stupid,  or  he  was  neither  mad,  miser- 
ably trained  nor  misled  by  bad  companions. 

The  application  of  the  term  contrary  to  complex  pro- 
positions is  of  less  interest.  We  may  however  consider  that 
we  have  the  contrary  of  such  a  proposition  when  we  deny 
every  part  of  the  statement.     Thus  the  contrary  of  '' X  is 

^  I  still  assume  that  the  subject  of  the  proposilion  is  a  singular  term. 


CHAP.  II.] 


PROPOSITIONS. 


65 


both  A  and  B''  is  ''X  is  neither  A  nor  B";  of  "X  is  A 
and  either  B  or  C,"  "Xis  neither  A,  B  nor  C";  and  of  the 
given  proposition,  "He  was  neither  stupid  nor  mad  nor 
miserably  trained  nor  misled  by  bad  companions." 

62.  What  is  the  precise  meaning  of  the  assertion 
that  a  proposition— say  "All  grasses  are  edible" — is 
false  ?        [Jevons,  Studies  in  Deductive  Logic,  p.  1 16.] 

Professor  Jevons  discusses  at  some  length  the  point  here 
raised,  but  I  find  myself  quite  unable  to  agree  with  what 
he  says  in  connection  with  it. 

He  commences  by  giving  an  answer,  which  may  be  called 
the  orthodox  one,  and  which  I  should  certainly  hold  to  be 
the  correct  one.    When  I  assert  that  a  proposition  is  false,  I 
mean  that  its  contradictory  is  true.     The  given  proposition 
is  of  the  form  A,  and  its  contradictory  is  the  corresponding 
O  proposition,— Some  grasses  are  not  edible.    When,  there- 
fore, I  say  that  it  is  false  that  all  grasses  are  edible,  I  mean 
that  some  grasses  are  not  edible.    Professor  Jevons  however 
continues,  "  But  it  does  not  seem  to  have  occurred  to  logi- 
cians in  general  to  inquire  how  far  similar  relations  could 
be  detected  in  the  case  of  disjunctive  and  other  more  com- 
plicated  kinds   of  propositions.      Take,  for  instance,    the 
assertion  that  *all  endogens  are  all  parallel-leaved  plants.' 
If  this  be  false,  what  is  true  ?    Apparently  that  one  or  more 
endogens  are  not  parallel-leaved  plants,  or  else  that  one  or 
more  parallel-leaved  plants  are  not  endogens.     But  it  may 
also  happen  that  no  endogen  is  a  parallel-leaved  plant  at 
all.     There  are  three  alternatives,  and  the  simple  falsity  of 
the  original  does  not  shew  which  of  the  possible   contra- 
dictories is  true." 

In  this  statement,  there  appear  to  me  to  be  two  errors. 
In  the  first  place,  in  saying  that  one  or  more  endogens  are 

K.  L.  2 


66 


PROPOSITIONS. 


[part  ir. 


not  parallel-leaved  plants,  we  do  not  mean  to  exclude  the 
possibility  that  no  endogen  is  a  parallel-leaved  plant  at  all. 
Symbolically,  Some  S  is  not  I^  does  not  exclude  No  S  is  P. 
The  three  alternatives  are  therefore  at  any  rate  reduced 
to  the  two  first  given.  But  in  the  second  place,  I  think 
Professor  Jevons  is  in  error  in  regarding  each  of  these 
alternatives  by  itself  as  a  contradictory  of  the  original 
proposition.  The  true  logical  contradictory  is  the  affirma- 
tion of  the  truth  of  one  or  other  of  these  alternatives.  If  the 
original  complex  proposition  is  false  we  certainly  know  that 
the  new  complex  proposition  limiting  us  to  such  alternatives 
is  true. 

The  point  at  issue  may  be  further  illustrated  by  taking 
the  proposition  in  question  in  a  symbolic  form.  Ail  S  is  ail 
P  is  a  complex  proposition,  resolvable  into  the  form.  Ail  S  is 
Fy  and  all  P  is  S.  In  my  view,  it  has  but  one  contradictory, 
namely,  EitJier  some  S  is  not  F,  or  some  F  is  not  S^  If  either 
of  these  alternatives  holds  good,  the  original  statement 
must  in  its  entirety  be  false ;  and  on  the  other  hand,  if  the 
latter  is  false,  one  at  least  of  these  alternatives  must  be 
true.  Professor  Jevons  speaks  as  if  Some  S  is  not  F  were 
by  itself  a  contradictory  of  All  S  is  all  F.  But  it  is  merely 
inconsistent  with  it.  They  may  both  be  false.  No  doubt 
in  ordinary  speech  contradictory  frequently  implies  no  more 
than  "  inconsistent  with,"  and  if  Professor  Jevons  means 
that  we  should  also  use  the  term  contradictory  in  this  sense 
in  Logic,  the  question  becomes  a  verbal  one.  But  he 
means  more  than  this;  he  seems  to  mean  that  in  some 
cases  we  can  find  no  proposition  that  must  be  true  when 
a  given  proposition  is  false.     And  here  I  hold  that  he  is 


wrong. 


'  The  contradictory  of"  All  »S"  is  all  /"'  may  also  be  expressed  ".S" 
and  P  are  not  coextensive." 


CHAP.  II.] 


PROPOSITIONS. 


67 


If  the  original  proposition  is  complex,  its  contradictory 
will  in  general  be  complex  too,  and  possibly  still  more 
complex ;  but  that  might  naturally  be  expected.  Compare 
the  two  preceding  sections,  where  several  cases  are  worked 
out  in  detail. 

The   above   will    I   think   indicate   how   misleading   is 
Professor  Jevons's  further   statement,— « It  will  be  shewn 
in  a  subsequent  chapter  that   a   proposition   of  moderate 
complexity  has  an  almost  unlimited  number  of  contradic- 
tory propositions,  which  are  more  or  less  in  conflict  with 
the  original.     The  truth  of  any  one  or  more  of  these  con- 
tradictories establishes  the  falsity  of  the   original,  but  the 
falsity  of  the  original  does  not  establish  the  truth  of  any 
one  or  more  of  its  contradictories."     No  doubt  a  complex 
proposition  may  yield  an  indefinite  number  of  other  propo- 
sitions the  truth  of  any  one  of  which  is  inconsistent  with  its 
own.     But  it  has  only  one  logical  contradictory,  which  con- 
tradictory as  suggested  above  is  likely  to   be  a  still  more 
complex  proposition  affirming  a  number  of  alternatives  one 
or  other  of  which  must  hold  if  the  original  proposition  is 
false. 

Wth  the  point  here  raised  Professor  Jevons  mixes  up 
another,  with  regard  to  whicJi  his  view  is  almost  more  mis- 
leadmg.     He  says,  ^' But  the  question  arises  whether  there 
is  not  confusion  of  ideas  in  the  usual  treatment  of  this 
ancient  doctrine  of  opposition,  and  whether  a  contradictory 
of  a  proposition  is  not  any  proposition  which  involves  the 
falsity  of  the  original,  but  is  not  the  sole  condition  of  it 
I  apprehend  that  any  assertion  is  false  which  is  made  with- 
out sufficient  grounds.     It  is  false  to  assert  that  the  hidden 
side  of  the  moon  is  covered  with  mountains,  not  because 
we  can  prove  the  contradictory,  but  because  we  know  that 
the  assertor  must  have  made  the  assertion  without  evidence. 

5—2 


68 


PROPOSITIONS. 


[part  II. 


If  a  person  ignorant  of  mathematics  were  to  assert  that 
*all  involutes  are  transcendental  curves/  he  would  be  making 
a  false  assertion,  because,  whether  they  are  so  or  not,  he 
cannot  know  it."  Surely  in  Logic  we  cannot  regard  the 
truth  or  falsity  of  a  proposition  as  depending  upon  the 
knowledge  of  the  person  who  affirms  it,  so  that  the  same 
proposition  would  now  be  true,  now  false.  The  question 
"What  is  truth?"  may  be  an  enormously  difficult  one  to 
answer  absolutely,  and  I  need  not  say  that  I  shall  not 
attempt  to  deal  with  it  here ;  but  unless  we  are  allowed  to 
proceed  from  the  falsity  of  "All  *$"  is  P''  to  the  truth  of 
"  Some  S  is  not  P,"  I  do  not  think  we  can  go  far  in  Logic. 

63.  Analyse  all  that  is  implied  in  the  assertion 
of  the  falsity  of  each  of  the  following  propositions  : — 

(i)     Ro^^^cr  Bacon  was  a  e^iant. 

(2) 

(3)  Bare  assertion  is  not  necessarily  the  naked 
truth. 

(4)  All  kinds  of  grasses  except  one  or  two 
species  are  not  poisonous. 

[Jevons,  Studies,  p.  124.] 

64.  Assign  precisely  the  meaning  of  the  assertion 
that  it  IS  false  to  say  that  some  English  soldiers  did 
not  behave  discreditably  in  South  Africa.  [l.] 

65.  Examine  in  the  case  of  each  of  the  follow- 
ing propositions  the  precise  meaning  of  the  assertion 
that  the  proposition  is  false : — 

(i)     Some  electricity  is  generated  by  friction. 


Descartes  died  before  Newton  was  born. 


CHAP.  II.] 


PROPOSITIONS. 


^ 


0*0  Oxygen  and  nitrogen  are  constituents  of 
the  air  we  breathe. 

(iii)  If  a  straight  line  falling  upon  two  other 
straight  lines  make  the  alternate  angles  equal  to 
each  other,  these  two  straight  lines  shall  be  parallel. 

(iv)     Actions  are  either  good,  bad,  or  indifferent. 


CHAPTER  III. 


THE   CONVERSION    OF   PROPOSITIONS. 


66.     The   meaning^  of    logical   conversion.      The 
ordinary  conversion  of  propositions. 

By  Conversion,  in  a  broad  sense,  is  meant  a  change  in 
the  position  of  the  terms  of  a  proposition  \ 

Logic,  however,  is  concerned  with  conversion  only  in 
so  far  as  the  truth  of  the  new  proposition  obtained  by  the 
process  is  a  legitimate  inference  from  the  truth  of  the 
original  proposition.  This  is  what  Whately  means  when 
he  says  that  "no  conversion  is  employed  for  any  logical 
purpose  unless  it  be  illative"  {Elements  of  Lo^^ic,  p.  74).  For 
example,  the  change  from  All  5  is  Z'  to  All  Z'  is  6"  is  not 
a  logical  conversion,  since  the  truth  of  the  latter  proposition 
does  not  follow  from  the  truth  of  the  former. 

The  simplest  form  of  logical  conversion  may  be  defined 
as  follows,  and  it  may  be  distinguished  from  other  forms  by 
being  called  ordinary  conversion  :— By  ordinary  conversion 
is  meant  a  process  of  immediate  inference  in  whieh  from  a 
given  proposition  we  infer  another,  having  the  predicate  of  the 
original  proposition  for  subject,  and  its  subject  for  predicate. 

^  Ueberweg  (Lindsay's  translation,  p.  -294)  defines  Conversion  thus. 
Compare  also  De  Morgan,  p.  58. 


CHAP.  III.] 


PROPOSITIONS. 


71 

Thus,  given  a  proposition  having  S  for  its  subject  and 
Piox  its  predicate,  we  seek  to  obtain  by  immediate  inference 
a  new  proposition  having  P  for  its  subject  and  6*  for  its 
predicate ;  and  applying  this  rule  to  the  four  fundamental 
forms  of  proposition,  we  get  the  following  table  :— 


Original  Proposition. 


Converse. 


All  5  is  P.     A. 


Some  6"  is  P.     I. 


No  S  is  P.     E. 


Some  5  is  not  P.    O. 


Some  P  is  S.    I. 


Some  P  is  S.     I. 


No  P  is  S.      E. 


(None.) 


67.  Simple  Conversion,  and  Conversion  per  ac- 
cidcns. 

It  will  be  observed  that  in  the  case  of  I  and  E,  the 
converse  is  of  exactly  the  same  form  as  the  original  pro- 
position (or  convertend) ;  we  do  not  lose  any  part  of  the 
information  given  us  by  the  convertend,  and  we  can  pass 
back  to  it  by  re-conversion  of  the  converse.  The  convertend 
and  its  converse  are  equivalent  propositions.  The  con- 
version in  both  these  cases  is  said  to  be  simple. 

In  the  case  of  A,  it  is  different;  although  we  start  with  a 
universal  proposition,  we  obtain  by  conversion  a  particular 
one  only,  and  by  no  means  of  operating  upon  the  converse 
can  we  regain  the  original  proposition.  The  convertend  and 
its  converse  are  not  equivalent  propositions.  This  is  called 
conversion /rr  accidens\  or  conversion  by  lijfiitatioji. 

^  The  conversion  of  A  is  said  by  Mansel  to  be  called  conversion 


72 


PROPOSITIONS. 


[part  II. 


68.  Particular  negative  propositions  do  not  admit 
of  ordinary  conversion. 

It  is  clear  that  if  the  converse  is  to  be  a  legitimate 
formal  inference  from  the  original  proposition  (or  convert- 
end),  it  must  distribute  no  term  that  was  not  distributed  in 
the  convertend.  From  this  it  follows  immediately  that 
Some  S  is  not  P  does  not  admit  of  ordinary  conversion; 
for  S  which  is  undistributed  in  the  convertend  would  be- 
come the  predicate  of  a  negative  proposition  in  the  converse, 
and  would  therefore  be  distributed.  (I  may  remind  the 
reader  that  in  what  I  have  called  ordinary  conversion, 
with  which  alone  we  are  now  dealing,  we  do  not  admit  the 
contradictory  of  either  the  original  subject  or  the  original 
predicate  as  one  of  the  terms  of  our  converse.) 

I  cannot  understand  why  Professor  Jevons  should  say 
that  the  fact  that  the  particular  negative  proposition  is  in- 
capable of  ordinary  conversion  "  constitutes  a  blot  in  the 
ancient  logic"  {Studies  in  Deductive  Logic,  p.  37).  Wq  shall 
find  subsequently  that  just  as  much  can  be  inferred  from 
the  particular  negative  as  from  the  particular  affirmative, 
(since  the  latter  unHke  the  former  does  not  admit  of  con- 
traposition). Less  can  be  inferred  from  either  of  them  than 
can  be  inferred  from  the  corresponding  universal  proposition, 
and  this  is  obviously  because  the  latter  gives  all  the  informa- 

pcr  accidens  '*  because  it  is  not  a  conversion  of  the  universal  per  st\ 
but  by  reason  of  its  containing  the  particular.  For  the  proposition 
*  Some  B\%A'  \% primarily  the  converse  of  *  Some  A  is  B,'  secondarily 
of  *  All  A  is  /?'"  (Mansel's  Aldrich,  p.  61).  Professor  Baynes  seems  to 
deny  that  this  is  the  correct  explanation  of  the  use  of  the  term  {Xao 
Analytic  of  Loi^ical  Forms,  p.  29)  ;  but  however  this  may  be,  I  do 
not  think  that  we  can  really  regard  the  converse  of  A  as  obtained 
through  its  subaltern.  We  proceed  directly  from  "All  A  is  B"  to 
"  Some  B  is  A  "  without  the  intervention  of  '*  Some  A  is  B.'' 


CHAP.  III.] 


PROPOSITIONS. 


73 


tion  given  by  the  particular  proposition  and  more  beside. 
No  logic,  symbolic  or  other,  can  actually  obtain  more  from 
the  given  information  than  the  ancient  logic  does. 

69.     Give    the   converse    of  the    following  pro- 
positions : — 

(i)  A  stitch  in  time  saves  nine. 

(2)  None  but  the  brave  deserve  the  fair. 

(3)  He  can't  be  wrong  whose  life  is  in  the  right. 

(4)  The  virtuous  alone  are  happy. 

No  difficulty  can  be  found  in  converting  or  performing 
other  immediate  inferences  upon  any  given  proposition  if 
it  is  once  brought  into  logical  form,  its  quantity  and  quality 
being  determined,  and  its  subject,  copula  and  predicate 
being  definitely  distinguished  from  one  another. 

If  this  rule  is  neglected,  the  most  absurd  results  may 
be  elicited.  For  example,  amongst  several  curious  converses 
of  the  first  of  the  above  propositions  I  have  had  seriously 
given,— Nine  stitches  save  a  stitch  in  time.  Here  it  is  of 
course  entirely  overlooked  that  ^'save"  cannot  be  a  logical 
copula.  The  proposition  may  be  written.  All  stitches  in 
time  I  are  |  things  that  save  nine  stitches.  This  being  an 
A  proposition  is  only  convertible  per  accidens,  thus,  Some 
things  that  save  nine  stitches  are  stitches  in  time.  The 
following  is  wrong,— The  means  of  saving  nine  stitches  is 
a  stitch  in  time;  since  there  may  be  other  ways  of  saving 
nine. 

*'  None  but  the  brave  deser\^e  the  fair."  For  the  converse 
of  this  I  have  had,— The  fair  deserve  none  but  the  brave ; 
and,  again,  No  one  ugly  deserves  the  brave.  Logically  the 
proposition  may  be  written.  No  one  who  is  not  brave  is 
deserving  of  the  fair.     This,  being  an  E  proposition,  may 


74 


PROPOSITIONS. 


[part  II. 


be  converted  simply,  giving,  No  one  deserving  of  the  fair 

is  not  brave. 

**  He  can't  be  wrong  whose  life  is  in  the  right."  Written 
in  strict  logical  form,  this  proposition  becomes,— No  one 
whose  life  is  in  the  right  is  able  to  be  in  the  wrong ;  and 
therefore  its  converse  is,— No  one  who  is  able  to  be  in  the 
wrong  is  one  whose  life  is  in  the  right.  This  proposition 
may  now  be  written  in  the  more  natural  but  not  strictly 
logical  form,  His  life  cannot  be  in  the  right  who  can  him- 
self be  wrong. 

"  The  virtuous  alone  are  happy."  In  logical  form  this 
may  be  written  either,  No  one  who  is  not  virtuous  is  happy, 
or  All  who  are  happy  are  virtuous.  Taking  it  in  the  first 
form,  the  converse  is— No  one  who  is  happy  is  not  virtuous ; 
and  from  this  we  may  again  get  the  second  form  by  changing 
its  quality' — All  who  are  happy  are  virtuous.  The  converse 
of  this  is, — Some  who  are  virtuous  are  happy. 

70.  State  in  logical  form  and  convert  the  follow- 
ing propositions : — ■ 

(i)  There's  not  a  joy  the  world  can  give  like 
that  it  takes  away. 

(2)  He  jests  at  scars  who  never  felt  a  wound. 

(3)  Axioms  arc  self-evident. 

(4)  Natives  alone  can  stand  the  climate  of  Africa. 

(5)  Not    one    of    the    Greeks    at    Thermopylae 

escaped. 

(6)  All  that  glitters  is  not  gold.  [c] 

71.  Give  all  the  logical  opposites  of  the  pro- 
position : — Some  rich  men  are  virtuous ;  and  also  the 


1  r 


Cf.  section  73. 


CHAP.  III.] 


PROPOSITIONS. 


75 


converse  of  the  contrary  of  its  contradictory.     How 
is  the  latter  directly  related  to  the  given  proposition  .'* 

72.  Point  out  any  possible  ambiguities  in  the 
following  propositions,  and  shew  the  importance  of 
clearing  up  such  ambiguities  for  logical  purposes : — 

(i)     Some  of  the  candidates  have  been  successful. 

(ii)     Either  some  gross  deception  was  practised 
or  the  doctrine  of  spiritualism  is  true, 
(iii)     All  are  not  happy  that  seem  so. 
(iv)     All  the  fish  weighed  five  pounds. 

Give  the  contradictory  and  (where  possible)  the 
converse  of  each  of  these  propositions. 


CHAPTER  IV. 


THE   OBVERSION   AND    CONTRAPOSITION    OF  PROPOSITIONS. 


73.     The  Obvcrsion  of  Propositions. 

Obversion  is  the  process  of  changing  the  quality  of  a  pro- 
position without  altering  its  meaning.  This  cliange  of  quahty 
may  ahvays  be  made  if  at  the  same  time  ive  substitute  for  the 
predicate  its  contradictory. 

Applying  this  rule,  \vc  have  the  following  table : — 


Original  Proposition. 

Obverse. 

All  ^  is  P.     A. 

No  S  is  not-P.     E. 

1 

Some  S  is  P.     I. 

No  5  is  P.     E. 

Some  iS  is  not  not-/*.    O. 

All  S  is  xioi-P.     A. 

Some  S  is  not  P.     O. 

Some  S  is  not-/*.     I. 

The  term  Obvcrsion  is  used  by  Professor  Bain,  and  it  is 
a  convenient  one.  The  process  is  also  called  Permutation 
(Fowler),  Aequipollcnce  (Ueberweg),  Infinitation  (Bo wen). 
Immediate  Inference  by  Privative  Conception  (Jevons),  Contra- 
version  (De  Morgan),  Contraposition  (Spalding). 


CHAP.  IV.] 


PROPOSITIONS. 


n 


Obversion  depends  on  the  supposition  that  two  negatives 
make  an  affirmative.  De  Morgan  {Formal  Logic,  pp.  3,  4) 
points  out  that  in  ordinary  speech  this  is  not  always  strictly 
true.  For  example,  "not  unable"  is  scarcely  used  as  strictly 
equivalent  to  *'  able,"  but  is  understood  to  imply  a  some- 
what lower  degree  of  ability.  "John  is  able  to  translate 
Virgil"  is  taken  to  mean  that  he  can  translate  it  well; 
"Thomas  is  not  unable  to  translate  Virgil"  is  taken  to 
mean  that  he  can  translate  it— indifferently.  71iis  distinc- 
tion, however,  depends  a  good  deal  on  the  accentuation  of 
the  sentence;  and  it  is  not  one  of  which  Logic  can  take 
account.  Logically,  "y^  "  and  "not  not-^  "  must  be  regarded 
as  strictly  equivalent. 

74.  Formal  Obvcrsion  distinguished  from  Mate- 
rial Obversion. 

By  Formal  Obversion  is  meant  the  kind  of  obversion 
discussed  in  the  previous  section,  and  I  think  that  it  is 
the  only  kind  of  obversion  that  Formal  Logic  need  re- 
cognise. 

Professor  Bain  uses  the  expression  Material  Obversion, 
and  by  it  he  means  the  process  of  making  "  Obverse  In- 
ferences which  are  justified  only  on  an  examination  of  the 
matter  of  the  proposition"  {Logic,  i.  p.  in).  He  gives  as 
examples,— "Warmth  is  agreeable;  therefore,  cold  is  dis- 
agreeable. War  is  productive  of  evil ;  therefore,  peace  Is 
productive  of  good.  Knowledge  is  good;  therefore,  igno- 
rance is  bad."  I  should  be  inclined  to  doubt  whether  these 
are  legitimate  inferences,  formal  or  otherwise.  The  con- 
clusions would  appear  to  require  quite  independent  investi- 
gations to  establish  them.  For  example,  granted  that  warmth 
is  agreeable,  it  might  be  that  every  other  state  of  temperature 
is  agreeable  also. 


78 


.  PROPOSITIONS. 


[part  II. 


75.  Give  the  obverse   of  the  following  proposi- 
tions : — 

(i)    Whatever  is,  is  right. 

(2)  No  news  is  good  news. 

(3)  Good  orators  are  not  always  good  statesmen. 

(4)  A  stitch  in  time  saves  nine. 

76.  Conversion  by  Contraposition. 

Contraposition  (also  called  Conversion  by  Negation)  is  a 
process  of  immediate  inference  in  which  from  a  given  proposi- 
tion we  infer  another  proposition  having  the  contradictory  of 
the  original  predicate  for  its  subject,  and  the  original  subject 
for  its  predicate^, 

'  There  is  some  clifTerence  between  logicians  as  to  whether  the  con- 
trapositive  of  All  6"  is  y  is  No  not-P  is  S  or  All  not-P  is  nof-S.  It  is 
merely  a  verbal  question,  depending  on  our  original  definition  of  contra- 
position. It  will  be  observed  that  All  not-/'  is  not-^"  is  the  obverse  of 
No  not-/' is  .S",  and  if  we  regard  All  not-/' is  not-6'asthe  contrapositivc 
of  All  6"  is  /',  our  definition  of  contraposition  must  be  altered  to — "a 
process  of  immediate  inference  in  which  from  a  given  proposition  we 
infer  another  proposition  having  the  contradictory  of  the  original  pre- 
dicate for  its  subject,  and  the  contradictory  of  the  original  subject  for  its 
pretlicate."  In  this  case,  what  I  have  originally  defined  as  contra- 
position may  be  called  conversion  by  negation.  Careful  note  should  be 
taken  of  this  difference  of  usage,  and  then  no  difficulty  is  likely  to 
result.  Taking  the  following  definition,  we  might  call  either  form  a 
contrapositivc  of  the  original  proposition, — "contraposition  is  a  process 
of  immediate  inference  in  which  from  a  given  proposition  we  infer  an- 
other proposition  having  the  contradictory  of  the  original  predicate  for 
its  subject."  It  is  here  left  an  open  question  whether  the  predicate  of 
the  contrapositivc  is  to  be  the  original  subject  or  the  contradictor)-  of 
the  original  subject. 

The  following  is  from  Mansel's  Aldrich^  p.  6r, — "Conversion  by 
contraposition,  which  is  not  employed  by  Aristotle,  is  given  by  Boethius 


CHAP.  IV.] 


PROPOSITIONS. 


79 


Thus,  given  a  proposition  having  S  for  its  subject  and  P 
for  its  predicate,  we  seek  to  obtain  by  immediate  inference 
a  new  proposition  having  not-P  for  its  subject  and  S  for  its 
predicate. 

From  the  definition  we  can  immediately  deduce  the 
following  rule  for  obtaining  the  contrapositivc  of  a  given 
proposition: — Obvcrt  the  original  proposition,  and  then  con- 
vert the  proposition  thus  obtained.  For  given  a  proposition 
with  S  for  subject  and  P  for  predicate,  ob version  will  yield 
a  new  proposition  with  »S  for  subject  and  not-/' for  predicate, 
and  the  conversion  of  this  will  make  not-jP  the  siibject  and 
6" the  predicate;  i.e.,  we  shall  have  found  the  contrapositivc 
of  the  given  proposition. 


in  his  first  book,  Dc  Syllogismo  Catcgorico.  He  is  followed  by  Petrus 
Ilispanus.  It  should  be  observed,  that  the  old  Logicians,  following 
Boethius,  maintain,  that  in  conversion  by  contraposition,  as  well  as 
in  the  others,  the  (juality  should  remain  unchanged.  Consequently  the 
converse  of  'AH  A  is  B"*  is  *A11  not-v?  is  no\.-A\  and  of  'Some  A  i.s 
not  /)','  'Some  not-7>  is  not  not-^.'  It  is  simpler,  however,  to  convert 
A  into  E  and  0  into  /  (*No  not-j5  is  .-/';  'Some  mA-B  is  A'')  as  is 
done  by  Wallis  and  Abp.  Whately  ;  and  before  Boethius  by  Apuleius 
and  Capella,  who  notice  the  conversion,  but  do  not  give  it  a  name. 
The  principle  of  this  conversion  may  be  found  in  Aristotle,  Top.  Ii.  8.  i, 
though  he  does  not  employ  it  for  logical  purposes." 

In  most  text-books,  no  definition  of  contraposition  is  given  at  all, 
and  it  may  be  pointed  out  that  in  the  attempt  to  generalise  from  special 
examj)les,  Jevons  in  his  Elementary  Lessons  in  Lo^ic  gels  into  difficulties. 
For  the  contrapositivc  of  A  he  gives  All  not-/*  is  not-^";  O  he  says  has 
no  contrapositivc,  (but  only  a  converse  by  negation,  Some  not-/*  is  S) ; 
and  for  the  contrapositivc  of  E  he  gives  No  /'  is  S.  I  have  failed  to 
discover  that  any  precise  meaning  can  be  attached  to  contraposition, 
according  to  which  these  results  are  obtainable. 

It  should  be  observed  that  if  in  contraposition  the  quality  of  the 
proposition  is  to  remain  unchanged  as  in  Jcvons's  contrapositivc  of  ^, 
then  the  contrapositivc  both  of  E  and  O  will  be  Some  not-/*  is  not 
not-.S', 


8o  PROPOSITIONS.  [part  ii. 

Applying  this  rule,  we  have  the  following  table: — 


Original  Proposition. 

Olroerse. 

Contrapositive. 

All  6- is  T'.   A. 

No  ^  is  not-/'.   E. 

No  not-/*  is  S.    E. 

Some  S  is  /*.    I. 

Some  S  is  not  not-/*.   0. 

(None.) 

No^Msy.   E. 

All  6"  is  not-/'.   A. 

Some  not-/*  is  S.  I. 

Some  .S"  is  not  P.   0. 

Some  S  is  not-/*.   I. 

Some  not-/*  is  S.   I. 

It  is  easy  to  shew  that  Some  S is  P  has  no  contrapositive; 
for  when  it  is  obverted,  it  becomes  a  particular  negative; 
but  particular  negatives  do  not  admit  of  ordinary  conver- 
sion, which  is  the  process  that  must  succeed  obversion  in 
order  that  a  contrapositive  may  be  arrived  at. 

It  may  be  helpful  if  we  here  sum  up  the  immediate 
inferences  that  have  been  obtained  up  to  this  point,  making 
use  of  the  symbols  explained  in  section  38,  and  denoting 
not-^by  S\no\.-Phy  P'-.— 


Original 
Proposition. 

Converse. 

Oh'ersc. 

Contrapositive^ . 

Oln'crted^ 
Contrapositive. 

SaP 

PIS        SeP' 

P'eS 

P'aS' 

SiP 

PiS        SoP' 

ScP 

PeS       SaP' 

P'iS 

P'oS' 

SoP 

SiP' 

ns 

P'oS' 

1  It  should  be  remembered,  as  explained  in  the  preceding  note, 


CHAP.  IV.] 


PROPOSITIONS. 


81 


It  will  be  shewn  presently  how  this  table  of  Immediate 
Inferences  may  be  expanded. 

With  regard  to  the  utility  of  the  investigation  as  to  what 
contrapositives  are  logically  inferrible  from  given  proposi- 
tions, the  following  may  be  quoted  from  De  Morgan: — 

"The  uneducated  acquire  easy  and  accurate  use  of  the 
very  simplest  cases  of  transformation  of  propositions  and  of 
syllogisms.  The  educated,  by  a  higher  kind  of  practice, 
arriv^e  at  equally  easy  and  accurate  use  of  some  more  com- 
plicated cases :  but  not  of  all  those  which  are  treated  in 
ordinary  logic.  Euclid  may  have  been  ignorant  of  the 
identity  of  'Every  ^is  K'  and  'Every  not-Fis  not-X,'  for 
anything  that  appears  in  his  writings :  he  makes  the  one 
follow  from  the  other  by  a  new  proof  each  time"  (Syllabus, 
P-  32). 

77.  How  is  Obversion  related  to  Conversion  by 
Negation  or  Contraposition  1 

Give  the  obverse  and  the  contrapositive  of  the 
following  propositions : — 

(a)     All  animals  feed  ; 

{b)     No  plants  feed  ; 

{c)     Only  animals  feed.  [l.] 

78.  Give  the  contrapositive  of  the  following  pro- 
positions : — 

(i)  A  stitch  in  time  saves  nine. 

(2)  None  but  the  brave  deserve  the  fair. 

(3)  He  can't  be  wrong  whose  life  is  in  the  right. 

(4)  The  virtuous  alone  are  happy. 

that  what  is  called  the  contrapositive  above  is  sometimes  called  the 
converse  by  negation,  and  what  is  called  the  obverted  contrapositive 
above  is  sometimes  simply  called  the  contrapositive. 

K.  L.  6 


82 


PROPOSITIONS. 


[part  II. 

79.  Explain  the  nature  of  Conversion  and  Con- 
traposition by  reference  to  the  following  proposi- 
tions:— 

All  associations  are  separable. 

There  are  volcanoes  which  are  never  at  rest,  [v.] 

80.  "  The  angles  at  the  base  of  an  isosceles 
triangle  are  equal." 

What  can  be  inferred  from  this  proposition  by 
Obversion,  Conversion,  and  Contraposition,  without 
any  appeal  to  geometrical  proof  .'^  [L.] 

81.  Transform  the  following  propositions  in  such 
a  way  that,  without  losing  any  of  their  force,  they 
may  all  have  the  same  subject  and  the  same  predi- 
cate : — 

No  not-P  is  S, 

All  P  is  not-5. 

Some  P  is  Sy 

Some  not-P  is  not  not-^. 

This  problem  may  be  briefly  solved  as  follows : — 

No  not-P  is  5-  No  5  is  not-P  =  All  S  is  P, 
All  P  is  not-S=  No  Pis  S        =  No  6*  is  P. 
Some  Pis  S    =  Some  S is  P. 
Some  not-P  is  not  not-6'=  Some  not-P  is  S 
=  Some  S  is  not-P  ::=  Some  S  is  not  P. 

82.  Describe  the  logical  relations,  if  any,  between 
each  of  the  following  propositions,  and  each  of  the 
others : — 

(i)  There  are  no  inorganic  substances  which  do 
not  contain  carbon ; 


CHAP.  IV.]  PROPOSITIONS.  33 

« 

(ii)     All  organic  substances  contain  carbon  ; 

(iii)     Some  substances  not  containing  carbon  are 


organic ; 


(iv)  Some  inorganic  substances  do  not  contain 
carbon.  [(jj 

This  question  can  be  most  satisfactorily  answered  by 
reducing  the  propositions  to  such  forms  that  they  all  have 
the  same  subject  and  the  same  i)redicate. 

83.  The  application  of  the  doctrines  of  Conver- 
sion and  Contraposition  to  Hypothetical  Propositions. 

In  a  hypothetical  proposition  the  antecedent  and  the 
consequent  correspond  respectively  to  the  subject  and  the 
predicate  of  a  categorical  proposition.  In  Conversion 
therefore  the  old  consequent  must  be  the  new  antecedent, 
and  in  Contraposition  the  denial  of  the  old  consequent 
must  be  the  new  antecedent.  Proceeding  as  before,  this 
gives  us  immediately  the  following  table : — 


Original  Proposition. 


UAhB,  ChD.  A. 


Converse. 


Con  traposidve. 


In  some  cases  in  which    If  C  is  not  Z>,  A  is 
C'\^  D,  A  is  B.    I.  not  B.    E. 


In  some  cases  in  which  In  some  cases  in  which 
A  is  B,  C  is  D.  I.         C  is  A  A  is  B.   I. 


If  A  is  B,  C  is  not 
Z>.    E. 


In  some  cases  in  which 
A  is  B,  C  is  not  D.  0. 


None. 


If  C  is  D,  A  is  not    In  some  cases  in  which 
B.    E.  C  is  not  /J,  A  is  B.  I. 


None. 


In  some  cases  in  which 
C  is  not  B>,  A  is  B.  I. 


6—2 


84 


PROPOSITIONS. 


[part  II. 


It  must  be  remembered  that  we  regard  the  quaUty  of 
a  hypothetical  proposition  as  determined  by  the  quahty  of 

the  conse([uent. 

The  obverse  of  a  hypothetical  proposition  is  usually 
awkward  to  express.  We  may  however  find  it  if  required ; 
e.g.,  the  obverse  of  "  If  ^  is  i?,  C  is  Z> "  is  *'  If  A  is  B,  C  is 
not  not-Z>." 

84.  Give  the  converse  and  the  contrapositive  of 
"  If  a  straight  line  falling  upon  two  other  straight 
lines  make  the  alternate  angles  equal  to  one  another, 
these  two  straight  lines  shall  be  parallel."  [l.] 

The  application  of  the  doctrines  of  Conversion  and 
Contraposition  to  Hypothetical  Propositions  may  be  illus- 
trated by  means  of  the  above  proposition.  We  must  note 
carefully  that  it  is  a  universal  affirmative,  and  is  therefore 
only  convertible  J>cr  accidcns.  This  is  a  point  particularly 
liable  to  be  overlooked  where  a  universal  converse  can  be 
legitimately  inferred  (as  in  the  case  of  the  above  proposition), 
thoudi  not  as  an  immediate  inference.  We  are  in  no  danger 
of  saying.  All  men  are  animals,  therefore,  all  animals  are 
men ;  but  we  may  be  in  danger  of  saying,  All  equilateral 
triangles  are  e(iuiangular,  therefore,  all  equiangular  triangles 
are  equilateral.  From  the  point  of  view  however  of  Formal 
Logic  the  latter  inference  is  as  erroneous  as  the  former. 

So  fiir  as  the  given  proposition  is  concerned,  wx  have— 
Converse,  In  some  cases  in  which  two  straight  lines  are 

parallel,  a  straight  line  falling  upon  them  shall  make  the 

alternate  angles  equal  to  one  another. 

Contrapositive,  If  two  straight  lines  are  not  parallel,  then 

a  straight  line  falling  upon  them  shall  make  the  alternate 

angles  not  equal  to  one  another. 


PROPOSITIONS. 


85 


CHAP.  IV.] 

85.  Give  the  contradictory,  the  contrary,  the 
converse,  and  the  contrapositive  of  the  following 
propositions : 

(i)  Things  equal  to  the  same  thing  arc  equal  to 
one  another. 

(2)  No  one  is  a  hero  to  his  valet. 

(3)  If  there  is  no  rain  the  harvest  is  never  good. 

(4)  None  think  the  great  unhappy  but  the  great. 

(5)  Fain  would  I  climb  but  that  I  fear  to  fall. 

86.  Name  the  form  of  each  of  the  following 
propositions  ;  and,  where  possible,  give  the  converse 
and  the  contrapositive  of  each : — 

(i)     Some  death  is  better  than  some  life, 
(ii)     The  candidates  in  each  class  arc  not  arranged 
in  order  of  merit. 

(iii)     Honesty  is  the  best  policy. 

(iv)     Not  all  that  tempts  your  wand'ring  eyes 
And  heedless  hearts  is  lawful  prize. 

(v)  If  an  import  duty  is  a  means  of  revenue,  it 
does  not  afford  protection. 

(vi)     Great  is  Diana  of  the  Ephesians. 

(vii)  All  these  claims  upon  my  time  overpower 
me. 


CHAPTER  V. 


THE    INVERSION    OF    PROPOSITIONS. 


87.  In  what  cases  can  we  obtain  by  immediate 
Inference  from  a  given  proposition  a  new  proposition 
havini^  the  contradictory  of  the  oric^inal  subject  for 
its  subject,  and  the  original  predicate  for  its  predi- 
cate ? 

A  new  form  of  immediate  inference  is  here  indicated, 
by  whicli  given  a  proposition  having  6"  for  its  subject  and 
/^  for  its  predicate,  we  seek  to  obtain  a  new  proposition 
having  not-.S'  for  its  subject  and  P  for  its  predicate. 

If  such  a  proposition  can  be  obtained  at  all,  it  will  be 
by  a  certain  combination  of  the  elementary  processes  of 
ordinary  conversion  and  obversion.  We  will  take  each 
of  the  fundamental  forms  of  i)roposition  and  see  what  can 
be  obtained  (i)  by  first  converting  it,  and  then  performing 
alternately  the  operations  of  obversion  and  conversion ; 
(2)  by  first  obverting  it,  and  then  performing  alternately 
the  operations  of  conversion  and  obversion.  We  shall  find 
that  in  each  case  we  can  go  on  till  we  reach  a  i)articular 
negative  proposition  whose  turn  it  is  to  be  converted. 

(i)     The  results  of  performing  the  processes  of  con- 
version and   obversion   alternately,   commencing   with  the 
former^  are  as  follows  : — 


PROPOSITIONS. 


^1 


CHAP,  v.] 

(i)     Alibis/', 

therefore  (by  conversion),  Some  P  is  S, 
therefore  (by  obversion),  Some  P  is  not  not-^l 

Here  comes  the  turn  for  conversion;  but  we  have  an 
O  proposition,  and  can  therefore  proceed  no  further. 

(ii)     Some  S  is  /*, 

therefore  (by  conversion).  Some  P  is  6*, 
therefore  (by  obversion),  Some  /'is  not  not-^*; 
and  we  can  get  no  further. 

(iii)     No  S  is  P, 

therefore  (by  conversion),  No  P  is  5, 
therefore  (by  obversion),  All  P  is  not-^, 
therefore  (by  conversion),  Some  iiot-S  is  P, 
therefore  (by  obversion).  Some  not-^"  is  not  not-/*. 

In  this  case  the  proposition  in  itaUcs  is  the  immediate 
inference  that  was  sought. 

(iv)     Some  S  is  not  P. 

In  this  case  we  are  not  able  even  to  commence  our 
series  of  operations. 

(2)  The  results  of  performing  the  processes  of  con- 
version and  obversion  alternately,  commencing  with  the 
latter^  are  as  follows  : — 

(i)     All  S  is  P, 

therefore  (by  obversion),  No  S  is  not-/*, 
therefore  (by  conversion).  No  not-/*  is  S, 
therefore  (by  obversion).  All  not-/*  is  not-.S', 
therefore  (by  conversion).  Some  not-^S  is  not-/*, 
therefore  (by  obversion),  So7ne  not-S  is  not  P. 

Here  again  we  have  obtained  the  desired  form. 

(ii)     Some  6*  is  /*, 

therefore  (by  obversion),  Some  S  is  not  not-/*. 


88 


PROPOSITIONS. 


[part  II. 


(iii)    NoSisPy 

therefore  (by  obversion),  All  S  is  not-P, 
therefore  (by  conversion),  Some  not-/'  is  S, 
therefore  (by  obversion),  Some  not-/*  is  not  not-S. 

(iv)     Some  »S  is  not  jP, 

therefore  (by  obversion).  Some  S  is  not-/*, 
therefore  (by  conversion).  Some  not-/*  is  S, 
therefore  (by  obversion),  Some  not-/*  is  not  not- 5. 
We  can  now  answer  the  question  with  which  we  com- 
menced this  enquiry.     The   required   proposition   can   be 
obtained  only  if  the  given  proposition  is  universal ;  we  then 
have,  according  as  it  is  affirmative  or  negative, — 

All  S  is  /*,  therefore.  Some  not-S  is  not  r ; 
No  S  is  Fy  therefore.  Some  not-*S*  is  /". 

It  must  be  observed  that  in  the  case  of  the  former  of 
these  we  commenced  with  obversion  in  order  to  get  the 
new  form,  in  the  latter  we  commenced  with  conversion. 

This  form  of  immediate  inference  has  been  more  or  less 
casually  recognised  by  various  logicians;  but  I  do  not 
remember  that  it  has  ever  received  any  distinctive  name. 
Sometimes  it  has  been  vaguely  classed  under  contraposition, 
(compare  Jevons,  Elementary  Lessons  in  LogiCy  pp.  185,  6), 
but  it  is  really  as  far  removed  from  the  process  to  which 
that  designation  has  been  given  as  the  latter  is  from  ordinary 
conversion.  I  venture  to  suggest  the  terms  Inversion  and 
Inverse',     Thus,  Inversion  is  a  proeess  of  immediate  inferenee 

1  For  assumptions  respecting  "existence"  involved  in  these  in- 
ferences, see  chapter  8. 

-  Professor  Jevons  (carrying  out  a  suggestion  of  Professor  Robert- 
son's) has  introduced  the  tonn  Inverse  in  a  different  sense.  I  do  not 
however  think  that  for  logical  purposes  we  want  any  new  term  in  the 
sense  in  which  he  uses  it  ;  ami  I  have  been  unable  to  think  of  any  other 
equally  suitable  term  for  my  own  purpose,  for  which  a  new  term  really 


CHAP,  v.] 


PROPOSITIONS. 


89 


in  whieh  front  a  given  proposition  we  infer  a7iot/ier  proposition 
/laving  the  eo?itradietory  of  the  original  subject  for  its  suhjecty 

is  needed,  if  the  scheme  of  immediate  inferences  by  means  of  conver- 
sion and  obversion  is  to  be  made  scientifically  complete.  The  term 
contrave^se  has  occurred  to  me,  but  I  do  not  like  it  so  well ;  and  this 
again  has  been  appropriated  by  De  Morgan  in  another  sense. 

Professor  Jevons's  nomenclature  is  explained  in  the  following  passage 
from  his  Studies  in  Deductive  Lo-^ic^  p.  32  : — "  It  appears  to  be  indis- 
j^ensable  to  endeavour  to  introduce  some  fixed  nomenclature  for  the 
relations  of  propositions  involving  two  terms.  Professor  Alexander 
liain  has  already  made  an  innovation  by  using  the  term  obverse,  and 
Professor  Hirst,  Professor  Ilenrici  and  other  reformers  of  the  teaching 
of  geometry  have  begun  to  use  the  terms  converse  and  obverse  in 
meanings  inconsistent  with  those  attached  to  them  in  logical  science 
[Mindy  1876,  p.  147).  It  seems  needful,  therefore,  to  state  in  the  most 
explicit  way  the  nomenclature  here  proposed  to  be  adopted  with  the 
concurrence  of  Professor  Robertson. 

Taking  as  the  original  proposition  'All  A  are  i?,'  the  following  are 
what  we  may  call  the  related  propositions — 

Inferrible. 
Converse.     Some  B  are  A. 
Obverse.     No  A  are  not  B. 
Contrapositive.     No  not  B  are  A,  or,  all  not  B  arc  not  A. 

Non-Inferrible. 

Inverse.     All  B  are  A. 

Reciprocal.     All  not  A  are  not  B. 

It  must  be  observed  that  the  converse,  obverse,  and  contrapositive 
are  all  true  if  the  original  proposition  is  true.  The  same  is  not  neces- 
sarily the  case  with  the  inverse  and  7'eciprocal.  These  latter  two  names 
are  adopted  from  the  excellent  work  of  Delbccuf,  rrolegomcnes  Philo- 
sophiqiies  de  la  Geometrie,  pp.  88 — 91,  at  the  suggestion  of  Professor 
Croom  Robertson  {Mind,  1876,  p.  425)." 

In  this  scheme  what  I  propose  to  call  the  Biverse  is  not  recognised 
at  all.  On  the  other  hand,  I  hardly  see  why  the  uon-infcrrible  forms 
need  such  a  distinct  logical  recognition  as  is  implied  by  giving  them 
distinct  names ;  while  except  in  books  on  Logic  I  anticipate  that  the 
term  converse  is  likely  still  to  be  used  in  its  non-logical  sense,  [i.e.,  *'A11 
B  are  ./"  is  likely  still  to  be  spoken  of  as  the  converse  of  "All  A  are 


90 


PROPOSITIONS. 


[part  II. 


and  the  ori^i?ial  predicate  for  its  predicate.  In  other  words, 
given  a  proposition  having  S  for  subject  and  F  for  predicate, 
we  obtain  by  inversion  a  new  proposition  having  not-^S  for 
subject  and  F  for  predicate. 

We  may  now  sum  up  the  results  that  have  been  obtained 
with  regard  to  immediate  inferences.  Given  two  terms  S 
and  /*,  and  admitting  their  contradictories  not-*S  and  not-/*, 
we  have  eight  possible  forms  of  proposition  as  shewn  in 
the  following  scheme: — 


Sidtjcct. 

Predicate* 
P 

(i) 

s 

(ii) 

s 

not-F 

1 

(iii) 

1 

p 

.S" 

(iv) 

p 

not-/* 

not-S 

(V) 

S 

(vi) 

not-/* 
not-^" 

noi-S 

(vii) 

P 

not-F 

'  (viii) 

not-^" 

/)  ").  It  may  be  noted  that  in  Jevons's  use  of  terms,  the  inverse  would 
be  the  same  as  the  converse  in  the  case  of  E  anil  I  propositions.  I 
imagine  also  that  in  consistency  there  should  be  yet  another  term  to 
express  the  relation  of  "No  not-/>'  is  not-A'^  or  "All  noi-B  is  ^4"  to 
"Xo  A  is  y>";  it  is,  in  the  sense  in  which  Jevons  uses  these  terms, 
neither  the  Converse,  Obverse,  Contrapositive,  Inverse  nor  Reciprocal. 


CHAP,  v.] 


PROPOSITIONS. 


91 


These  propositions  may  be  designated  respectively: — 

(i)     The  original  proposition, 

(ii)    The  obverse, 

(iii)   The  converse, 

(iv)    The  obverted  converse, 

(v)    The  contrapositive, 

(vi)    The  obverted  contrapositive, 

(vii)   The  inverse, 

(viii)  The  obverted  inverse. 

It  has  been  shewn,  in  sections  66,  73,  76,  and  in  the 
above,  that  if  the  original  proposition  is  universal,  we  can 
infer  from  it  propositions  of  all  the  remaining  seven  forms ; 
but  if  it  is  particular,  we  can  infer  only  three  others. 

Working  out  the  different  cases  in  detail  we  have: — 

A.     (i)  Original  proposition,  A/l  S  is  F. 

(ii)  Obverse,  M?  S  is  not-F. 

(iii)  Converse,  Some  F  is  S. 

(iv)  Obverted  converse,  Some  F  is  not  not-S. 

(v)  Contrapositive,  No  not-F  is  S. 

(vi)  Obverted  Contrapositive,  All  not-F  is  not-S. 

(vii)  Inverse,  Some  not-S  is  not  F. 

(viii)  Obverted  Inverse,  Some  not-S  is  not-F. 

I.      (i)  Original  proposition,  Some  S  is  F. 

(ii)  Obverse,  Some  S  is  not  not-F. 

(iii)  Converse,  Some  F  is  S. 

(iv)  Obverted  Converse,  Some  F  is  not  not-S. 

(v)  Contrapositive,  none  can  be  inferred, 

(vi)  Obverted  Contrapositive,  none, 

(vii)  Inverse,  none, 

(viii)  Obverted  Inverse,  none. 

E.    (i)      Original  proposition,  No  S  is  F, 
(ii)     Obverse,  All  S  is  not-F. 


93 


PROPOSITIONS. 


[part  II. 


(iii)    Converse,  No  P  is  S. 
(iv)    Obverted  Converse,  All  P  is  ?iot-S. 
(v)     Contrapositive,  Sovie  not-P  is  S. 
(vi)    Obverted  Contrapositive,  Some  not-P  is  not  fiot-S. 
(vii)    Inverse,  Some  not-S  is  P. 
(viii)    Obverted  Inverse,  Some  not-S  is  not  not-P. 

O.     (i)      Original  proposition,  Some  S  is  not  P. 
(ii)     Obverse,  Some  S  is  not-P. 
(iii)     Converse,  none  can  be  inferred, 
(iv)    Obverted  Converse,  none, 
(v)     Contrapositive,  Some  not-P  is  S. 
(vi)    01)verted  Contrapositive,  Some  not-P  is  not  not-S. 
(vii)    Inverse,  none, 
(viii)  Obverted  Inverse,  none. 
AH  the  above  is  summed  ii})    in   the   following   Table 
(using  the  symbols  described  in  section   '^^'^^  and  denotin 
not-i  by  S'.xioi-P  by  P')\— 


o 


I. 


Original  I'roiK)sition 


SaP        SiP     \    ScP    I    SoP 


11 


111 


Obverse    '    ScP'       SoP'    ,    SaP'       SiP' 


Converse 


PiS        PiS     '    PcS 


iv 

Obverted  Converse 

v 

ContraDositive 

vi 

Obverted  Contrapositive. . . 

PoS'       PoS' 


t  .  v 


7\uS 


P\S 
P'aS' 


P'/S       P'iS    j 


vu 


Inverse 


S'oP 


via 


Obverted  Inverse S'iP' 


P'oS'      P'oS' 

S'iP 

S'oP' 

\ 

! 

CHAP,  v.] 


PROPOSITIONS. 


93 


It  is  worth  noticing  that  we  can  infer  the  same  number 
of  propositions  from  E  as  from  A  (7),  from  O  as  from  I  (3), 
and  the  same  number  of  universal  propositions  from  E  as 
from  A  (3);  also  in  two  cases  we  can  get  no  more  from  A 
than  from  I,  and  no  more  from  E  than  from  O. 

88.  Give  the  inverse  of  the  following  proposi- 
tions : — 

• 

(i)  A  stitch  in  time  saves  nine. 

(2)  None  but  the  brave  deserve  the  fair. 

(3)  He  can't  be  wrong  whose  life  is  in  the  right. 

(4)  The  virtuous  alone  are  happy. 

89.  Assuming  that  no  organic  beings  arc  devoid 
of  Carbon,  what  can  we  thence  infer  respectively 
about  beings  which  are  not  organic,  and  things  which 
arc  not  devoid  of  carbon  .'*  [l.] 

90.  Make  as  many  Immediate  Inferences  as  you 
can  from  the  following  propositions  : — 

(i)     Civilization  and  Christianity  are  coextensive. 

(2)  Uneasy  lies  the  head  that  wears  a  crown. 

(3)  Your  money  or  your  life  !  [l.] 

91.  Write  out  all  the  propositions  that  must  be 
true,  and  all  that  must  be  false,  if  we  grant  that 

(a)  A  straight  line  Is  the  shortest  distance  be- 
tween two  points ; 

(y3)  All  the  angles  of  a  triangle  are  equal  to  two 
right  angles ; 

(7)     Not  all  the  great  are  happy.  [c] 


94 


PROPOSITIONS. 


[part  II. 


92.  De  Morgan  says  {Fourth  Memoir  on  the 
Syllogism,  p.  5)  of  the  Laws  of  Thought :  "  Every 
transgression  of  these  laws  is  an  invahd  inference ; 
every  valid  inference  is  not  a  transgression  of  these 
laws.  But  I  cannot  admit  that  everything  which  is 
not  a  transgression  of  these  laws  is  a  valid  inference." 
Investicrate  the  locrical  relations  between  these  three 


assertions. 


[Jevons,  Studies,  p.  301.] 


93.  Assign  the  logical  relation,  if  any,  between 
each  pair  of  the  following  propositions  : — 

(i)  All  crystals  are  solids. 

(2)  Some  solids  are  not  crystals. 

(3)  Some  not  crystals  are  not  solids. 

(4)  No  crystals  are  not  solids. 

(5)  Some  solids  are  crystals. 

(6)  Some  not  solids  are  not  crystals. 

(7)  All  solids  are  crystals.  [l.] 


94.     "All  that  love  virtue  love  angling." 

Arrange  the   following   propositions   in  the  four 
following  groups : — 

(a)     Those  which  can  be  inferred  from  the  above 
proposition ; 

(/3)    Those  from  which  it  can  be  inferred  ; 

(7)  Those  which  do  not  contradict  it,  but  which 
cannot  be  inferred  from  it ; 

(8)  Those  which  contradict  it. 


CHAP,  v.]  PROPOSITIONS. 

(i)  None  that  love  not  virtue  love  andine. 

(ii)  All  that  love  angling  love  virtue, 

(iii)  All  that  love  not  angling  love  virtue, 

(iv)  None  that  love  not  angling  love  virtue. 

(v)  Some  that  love  not  virtue  love  angling, 

(vi)  Some  that  love  not  virtue  love  not  ancflincr. 

(vii)  Some  that  love  not  angling  love  virtue, 

(viii)  Some  that  love  not  angling  love  not  virtue. 


95 


CHAPTER  VI. 


THE    DIAGRAMMATIC    REPRESENTATION    OF    PROPOSITIONS. 


95.  Methods  of  illustrating^  the  ordinary  processes 
of  Formal  Logic  by  means  of  Diagrams. 

Representing  the  individuals  included  in  any  class,  or 
denoted  by  any  name,  by  a  circle,  it  will  be  obvious  that 
the  five  following  diagrams  represent  all  possible  relations 
between  any  two  classes : — - 


The  force  of  the  different  propositional  forms  is  to  ex- 
clude one  or  more  of  these  possibilities'. 

^  The  method  of  interpreting  a  proposition  by  what  it  excludes  or 
negatives  is  discussed  in  more  detail  in  chapter  vill. 


CHAP.  VI.]  PROPOSITIONS. 

All  S  is  P  limits  us  to 


97 


or 


Some  S  is  P  to  one  of  the  four 


H- 


No  S  is  P  to 


Soffie  S  is  not  P  to  one  of  the  three 


QB- 


To  represent  All  6"  is  /*  by  a  single  diagram,  thus 


K.  L. 


98  PROPOSITIONS, 

or  Some  Sis  Phy  a  single  diagram,  thus 


[part  II. 


is  most  misleading;  since  in  each  case  the  proposition  really 
leaves  us  with  other  alternatives.  This  method  of  employ- 
ing the  diagrams  is  however  adopted  by  most  logicians  who 
have  used  them,  including  Sir  William  Hamilton  {Logic,  i. 
p.  255),  and  Professor  Jevons  {Elementary  Lessons  in  Logic ^ 
pp.  72 — 75);  and  the  attempt  at  such  simplification  has 
brought  their  use  into  undeserved  disrepute.  Thus,  Mr 
Venn  remarks,  "The  common  practice,  adopted  in  so  many 
manuals,  of  appealing  to  these  diagmms, — Eulerian  diagrams 
as  they  are  often  called, — seems  to  me  very  questionable. 
The  old  four  propositions  A,  E,  I,  O,  do  not  exactly  corre- 
spond to  the  five  diagrams,  and  consequently  none  of  the 
moods  in  the  syllogism  can  in  strict  propriety  be  represented 
by  these  diagrams?  {Symbolic  L.ogic,  p.  15,  compare  also  pp. 
424,  425).  This  is  undoubtedly  sound  as  against  the  use 
of  Euler's  circles  by  Hamilton  and  Jevons;  but  I  do  not 
admit  its  force  as  against  their  use  in  the  manner  described 
above  \  Many  of  the  operations  of  Formal  Logic  can  be 
satisfactorily  illustrated  by  their  aid;  though  it  is  true  that 
they  become  somewhat  cumbrous  in  relation  to  the  Syllogism. 
Thus,  they  may  be  employed, — (i)  To  illustrate  the  distri- 
bution of  the  predicate  in  a  proposition.  In  the  case  of  each 
of  the  four  fundamental  propositions  we  may  shade  the  part 
of  tlie  predicate  concerning  which  knowledge  is  given  us. 
"We  then  have, — 

> 

^  They  are  used  correctly  by  Ueberweg.     Cf.  Lindsay's  translation 
of  l^eberweg's  System  of  Logic,  pp.  216 — 218. 


CHAP.  VI.] 


PROPOSITIONS. 


99 


A. 


E. 


I. 


O. 


The  result  is  that  with  A  and  I  there  are  cases  in  which 
only  part  of /'is  shaded;  whereas  with  E  and  O,  the  whole 
of  F  is  in  every  case  shaded;  and  it  is  made  clear  that 
negative  propositions  distribute,  while  affirmative  proposi- 
tions do  not  distribute  their  predicates. 

(2)  To  illustrate  the  Opposition  of  Propositions.  Com- 
paring two  contradictory  propositions,  e.g.,  A  and  O,  we  see 
that  they  have  no  case  in  common,  but  that  between  them 
they  exhaust  all  possible  cases.  Hence  the  truth,  that  two 
contradictory  propositions  cannot  be  true  together  but  that 
one  of  them  must  be  true,  is  brought  home  to  us  under  a 
new  aspect.  Again,  comparing  two  subaltern  propositions, 
e.g.,  A  and  I,  we  notice  that  the  former  gives  us  all  the 
information  given  by  the  latter  and  something  more,  since 
it  still  further  limits  the  possibilities. 

To  make  this  point  the  more  clear  the  following  table  is 
appended : — 

7—2 


CHAP.  VI.] 


PROPOSITIONS. 


lOI 


_  (3)  To  illustrate  the  Conversion  of  Propositions.  Thus, 
it  is  made  quite  clear  how  it  is  that  A  admits  only  of  Con- 
version per  accidens.     All  S  is  P  limits  us  to  one  or  other  of 


the  following 


©■ 


The  problem  of  Conversion  is— What  do  we  know  of  P  in 
either  case?  In  the  first,  we  have  All  P  is  S,  but  in  the 
second  Some  P  is  S\  />.,  taking  the  cases  indifferently,  we 
have  Some  Ph  S and  nothing  more. 

Again,  it  is  made  clear  how  it  is  that  O  is  inconvertible. 
Some  S  is  not  P  limits  us  to  one  or  other  of  the  following, 


What  then  do  we  know  concerning  P.?   The  three  cases  give 
us  respectively 

(i)     NoPis^, 

(ii)     Some  P  is  5,  and  Some  P  is  not  S, 

(iii)    All  Pis  ^. 

(i)  and  (iii)  are  contraries,  and  (ii)  is  contradictory  to 
both  of  them.  Hence  nothing  can  be  affirmed  of  P  that  is 
true  in  all  three  cases  indifferently. 

(4)  To  illustrate  the  more  complicated  forms  of  imme- 
diate inference.  Taking,  for  example,  the  proposition  All 
S  is  P,  we  may  ask,  What  does  this  enable  us  to  assert  about 


102 


PROPOSITIONS. 


[part  II. 


not-P  and  not-S  respectively?     We  have  one  or  other  of 
these  cases 


W' 


With  regard  to  not-/*,  these  yield  respectively, 

(i)     No  not-jR  is  S, 

(ii)  No  not-/*  is  S,  And  thus  we  obtain  the  contraposi- 
tive  of  the  given  proposition. 

With  regard  to  not-^"  we  have 
(i)     All  not-S  is  not-/', 

(ii)  Some  not-^"  is  not-/*,  (unless  /*  constitutes  the 
entire  universe  of  discourse,  a  point  that  is  further  discussed 
subse(iuently);  />.,  in  either  case  we  may  infer  Some  not-^" 
is  not-/*.         E,  I,  O  may  be  dealt  with  similarly. 

The  application  of  the  diagrams  to  syllogisms  and  to 
special  problems  will  be  shewn  in  subsequent  sections. 

With  regard  to  all  the  above,  it  may  be  said  that  the  use 
of  the  circles  gives  us  nothing  that  could  not  easily  have 
been  obtained  independently.  This  is  of  course  true;  but 
no  one,  who  has  had  experience  of  the  difficulty  that  is 
sometimes  found  by  students  in  really  mastering  the  ele- 
mentary principles  of  Formal  Logic,  and  especially  in  deal- 
ing with  immediate  inferences,  will  despise  any  means  of 
illustrating  afresh  the  old  truths,  and  presenting  them  under 
a  new  aspect. 

The  fiict  that  we  have  not  a  single  diagram  correspond- 
ing to  each  fundamental  form  of  proposition  is  fatal  if  we 
wish  to  illustrate  any  complicated  train  of  reasoning  in  this 
way ;  but  in  indicating  the  real  nature  of  the  knowledge 


CHAP.  VI.] 


PROPOSITIONS. 


103 


given  by  the  propositions  themselves,  it  is  rather  an  advan- 
tage as  shewing  how  limited  in  some  cases  this  knowledge 
actually  is. 

The  diagrams  invented  and  used  by  Mr  Venn  {Symbolic 
LogiCy  Chapter  5)  are  extremely  interesting  and  valuable. 
In  this  scheme  the  diagram 


<3D' 


does  not  itself  represent  any  proposition,  but  the  framework 
into  which  propositions  may  be  fitted.  Denoting  not-S  by 
S'  and  what  is  both  S  and  F  by  SP,  &c.,  it  is  clear  that 
everything  must  be  contained  in  one  or  more  of  the  four 
classes  SP,  SP\  SP,  S'P'',  and  the  above  diagram  shews 
four  compartments,  (one  being  that  which  lies  outside  both 
the  circles),  corresponding  to  these  four  classes.  Every 
universal  proposition  denies  the  existence  of  one  or  more 
of  such  classes,  and  it  may  therefore  be  diagrammatically  re- 
presented by  shading  out  the  corresponding  compartment 
or  compartments.  Thus,  A/l  S  is  /*,  which  denies  the  exis- 
tence of  SP',  is  represented  by 


No  S  is  P  by 


P  . 


104  PROPOSITIONS.  [part  ii. 

If  we  have  three  terms  we  have  three  circles  and  eight 
compartments,  thus: — 


All  S  is  P  or  C  is  represented  by 


All  S  is  P  and  Q  by 


It  IS  in  cases  Involving  three  or  more  terms  that  the 
advantage  of  this  scheme  over  the  Eulerian  scheme  is  most 
manifest.  It  is  not  however  so  easy  to  apply  these  diagrams 
to  the  case  of  particular  propositions  \ 

'  "  If  we  introduce  particular  propositions  we  must  of  course  employ 

some  additional   form   of  dwgrammatical  notation We  might,  for 

example,  just  draw  a  bar' across  the  compartments  declared  to  be  saved; 
remembering  of  course  that,  whereas  destruction  is  distributive,  z.^.. 


CHAP.  VI.] 


PROPOSITIONS. 


105 


Lambert's  scheme  of  representing  propositions  by  com- 
binations of  straight  lines  will  be  touched  upon  in  connection 
with  the  syllogism ;  compare  section  1 80. 

[A  passing  reference  may  be  made  to  the  fundamental 
objection  raised  by  Mansel  against  the  introduction  of  any 
such  aids  at  all.  "  If  Logic  is  exclusively  concerned  with 
Thought,  and  Thought  Is  exclusively  concerned  with  Con- 
cepts, it  is  impossible  to  approve  of  a  practice,  sanctioned 
by  some  eminent  Logicians,  of  representing  the  relation  of 
terms  In  a  syllogism  by  that  of  figures  In  a  diagram.  To 
illustrate,  for  example,  the  position  of  the  terms  in  Barbara, 
by  a  diagram  of  three  circles,  one  within  another,  is  to  lose 
sight  of  the  distinctive  mark  of  a  concept,  that  it  cannot 
be  presented  to  the  sense,  and  tends  to  confuse  the  mental 
inclusion  of  one  notion  In  the  sphere  of  another,  with  the 
local  inclusion  of  a  smaller  portion  of  space  in  a  larger" 
{Prolegomena  Logica,  p.  55).  In  answering  this  objection, 
it  seems  sufficient  to  point  out  that  even  conceptualist 
logicians  must  recognise  and  deal  with  the  extension  of 
concepts,  and  that  the  Eulerian  diagrams  make  no  pretence 
of  representing  the  concepts  themselves,  but  only  their 
extension.] 

96.  Illustrate  the  relation  between  A  and  E,  and 
between  I  and  O  by  means  of  the  Eulerian  diagrams. 

cz'cry  included  sub-section  is  destroyed,  the  salvation  is  only  alternative 
or  partial,  i.e.,  we  can  only  be  sure  that  some  of  the  included  sub-sec- 
tions are  saved.  Thus,  '  No  x  is  j^,'  leading  to  destruction  of  xy,  will 
destroy  both  xyz  and  xy'z'\  [z  denoting  not-c),  "if  s  has  to  be  taken 
account  of.     IJut  '  Some  x  is  j',  saving  a  part  of  xy,  does  not  in  the 

least  indicate  whether  such  part  is  xyz  or  xyz If  it  were  worth  while 

thus  to  illustrate  complicated  groups  of  propositions  of  the  kind  in 
question,  it  could,  I  fancy,  be  done  with  very  tolerable  success."  Venn 
in  Mindy  1883,  pp.  599,  600. 


io6 


PROPOSITIONS. 


[part  II. 


97.  Illustrate  the  conversion  of  I,  the  contra- 
position of  O,  and  the  inversion  of  E,  by  means  of 
the  Eulerian  diagrams. 

98.  To  what  extent,  if  any,  may  the  processes 
of  Immediate  Inference  be  illustrated  by  means  of 
Mr  Venn's  diagrams  } 

99.  Any  information  given  with  respect  to  two 
terms  limits  the  possible  relations  between  them  to 
one  or  more  of  the  five  following, — 

0, 


Shew  how  such  information  may  in  all  cases  be  ex- 
pressed by  means  of  the  propositional  forms  A,  I,  E,  O. 

Let  the  five  relations  be  designated  respectively  a,  p, 
7,  S,  c'.  Information  is  given  when  the  possibility  of  one 
or  more  of  these  is  denied;  in  other  words,  when  we  are 
limited  to  one,  two,  three,  or  four  of  them.     Let  limitation 

^  Thus,  the  terms  being  S  and  P,  a  denotes  that  S  and  P  are 
wholly  coincident ;  /3  that  P  contains  S  and  more  besides  ;  y  that  ^ 
contains  P  and  more  besides  ;  5  that  S  and  P  overlap  each  other,  but 
that  each  includes  something  not  included  by  the  other  ;  e  that  S  and  P 
have  nothing  whatever  in  common. 


CHAP.  VI.] 


PROPOSITIONS. 


107 


to  a  or  /?,  (i.e.,  the  exclusion  of  y,  S  and  e),  be  denoted  by 
a,  /?;  limitation  to  a,  P  or  y,  (i.e.,  the  exclusion  of  S  and  e), 
by  a,  )8,  y;  and  so  on. 

Now  if  we  wish  to  express  such  information  by  means 
of  the  four  ordinary  propositional  forms,  we  find  that  some- 
times a  single  proposition  will  suffice  for  our  purpose;  thus 
a,  P  is  expressed  by  "  All  S  is  P."  Somedmes  we  require 
a  combination  of  propositions;  thus  a  is  expressed  by  saying 
that  "All  S  is  F,  and  also  All  F  is  5,"  (since  All  S  isF 
excludes  y,  8,  e,  and  All  F  is  S  further  excludes  p).  Some 
other  cases  are  still  more  complicated;  thus  the  fact  that 
we  are  limited  to  a  or  S  cannot  be  expressed  more  simply 
than  by  saying  "  Either  All  S  k  F  and  All  F  is  S,  or  else 
Some  S  is  F,  Some  S  is  not  F,  and  Some  F  is  not  5." 

Let  A  =  All  5  is  Fy  A,  =  All  F  is  S,  and  similarly  for 
the  other  proposidons.  Also  let  A Ai  =  All  S  is  F  and  All 
F  is  S,  &c.  Then  the  following  is  a  scheme  for  all  possible 
cases : — 


)o8 


PROPOSITIONS. 


[part  II. 


limitation  to 

denoted  by 

j  limitation  to 

denoted  by 

a 

AA, 

«,A7 

A  or  A, 

P 

AO, 

I     a,  AS 

A  or  lO, 

y 

A.O 
100, 

«,  Ac 

A  or  E 

s 

;    «» 7»  S 

A,  or  lO 

€ 

£ 

a»  y»  € 

A,  or  E 

°, /3 

A 

a,  S,  € 

AA,  or  OO, 

a,  8 

A, 

Ay,  8 

lO  or  lO, 

AA,  or  lOO, 

AA,  or  E     j 

i    Ay,  c 

AO,  or  A,0  or  E 

a,  £ 

j    A«,  € 

o, 

Ay 

AO,  or  A^O 

1 

y,  K  € 
«,  /?,  y,  8 

o 

AS 

lO, 

I 

A« 

AO,  or  E    ! 

1 

o,  ^,  y,  c 

A  or  A,  or  E 

7.8 

lO 

a,  /3,  8,  € 

AorO, 

7>  < 

A,0  or  E 

a,  y,  8,  € 

, 

A,  or  O 

S,  * 

OO, 

A  y,  8,  € 

O  or  O, 

It  will  be  found  that  any  other  combinations  of  pro- 
positions than  those  given  here  involve  either  contradictions 
or  redundancies,  or  else  no  information  is  given  because 
all  the  five  relations  that  are  a  priori  possible  still  remain 
possible. 

For  example,  AI  is  clearly  redundant ;  AO  is  self-con- 
tradictory;  A  or  A^O  is  redundant  (since  the  same  informa- 


CHAP.  VI.] 


PROPOSITIONS. 


109 


tion  is  given  by  A  or  A,);  A  or  O  gives  no  information 
(since  it  excludes  no  possible  case).  The  student  is  recom- 
mended to  test  other  combinations  similarly.  It  must  be  re- 
membered that  I,  =  I  and  E,  =  E. 

It  should  be  noticed  that  if  we  read  the  first  column 
downwards  and  the  second  column  upwards  we  get  pairs 
of  contradictories. 


CHAPTER  VII. 


THE    LOGICAL   FOUNDATION    OF   IMMEDIATE   INFERENCES. 


100.  Attempts  to  reduce  immediate  inferences 
to  the  mediate  form\ 

Immediate  inference  is  usually  defined  as  the  inference 
of  a  proposition  from  a  single  other  proposition ;  whereas 
mediate  inference  is  the  inference  of  a  preposition  from  at 
least  two  other  propositions. 

(i)  One  of  the  old  Greek  Logicians,  Alexander  of 
Aphrodisias,  establishes  the  conversion  of  E  by  means  of 
a  syllogism  in  Ferio. 

No  S  is  P, 

therefore,  No  -P  is  6"; 

for  if  not,  then  by  the  law  of  contradiction,  Some  P  is  S) 
and  we  have  this  syllogism, — 

No  S  is  P, 
Some  P  is  6", 

tlierefore,  Some  P  is  not  P, 
a  rcduitio  ad  absiirdutn, 

^  Students  who  have  not  already  a  technical  knowledge  of  the 
syllogism  may  omit  this  section  until  they  have  read  tlie  earlier  chapters 
of  Part  III. 


CHAP.  VII.] 


PROPOSITIONS. 


Ill 


Having  proved  the  conversion  of  E,  those  of  A  and  I 
will  follow  from  it. 

All  S\?>P,  ^ 

therefore,  Some  P'\^  S) 
for,  if  not,  No/^is  S\ 
and  therefore  (by  conversion)  No  *Sis  Z'; 
but  this  is  inconsistent  with  the  original  supposition. 
Similarly  for  I.     (Compare  Hansel's  Aldrich,  p.  62.) 

(2)  The  contraposition  of  A  may  be  established  by 
means  of  a  syllogism  in  Games t res  as  follows, — 

Given  All  ShP, 

we  have  also       No  not-P  is  P,  by  the  law  of  contradiction, 
therefore,  No  not-P  is  S. 

(3)  There  is  likewise  an  implicit  syllogism  in  the 
following  from  Jevons,  Studies  in  Deductive  Logic,  p.  44, 
"We  may  also  prove  the  truth  of  the  contrapositive  (of  the 
proposition  All  X  is  Y)  indirectly ;  for  what  is  not-K  must 
be  either  X  or  not-X;  but  if  it  be  X  it  is  by  the  premiss 
also  F,  so  that  the  same  thing  would  be  at  the  same  time 
not-F  and  also  F,  which  is  impossible.  It  follows  that  we 
must  affirm  of  not-F  the  other  alternative,  not-X" 

All  the  above  are  interesting,  as  illustrating  the  nature 
of  immediate  inferences ;  but  regarded  as  proofs  they  labour 
under  the  disadvantage  of  deducing  the  less  complex  by 
means  of  the  more  complex. 

I  hardly  know  what  is  to  be  said  in  favour  of  the  follow- 
ing :— 

(4)  AVolf  obtains  the  subaltern  of  a  universal  proposition 
by  a  syllogism  in  Darii, 


^  This  is  itself  an  inference  by  Opposition. 


112 

Given 
we  have  also 


PROPOSITIONS. 


[part  II. 


All  S  is  P, 

Some  S  is  6",  by  the  law  of  Identity, 

therefore,  Some  S  is  /*. 
(Compare,  Mansel,  Prolegomena  Logica,  p.  217.) 

(5)  "Still  more  absurd  is  the  elaborate  system  which 
Krug,  after  a  hint  from  Wolf,  has  constructed,  in  which  all 
immediate  inferences  appear  as  hypothetical  syllogisms; 
a  major  premiss  being  supplied  in  the  form,  *  If  all  A  is  B^ 
some  A  is  B.^  The  author  appears  to  have  forgotten,  that 
either  this  premiss  is  an  additional  empirical  truth,  in  which 
case  the  immediate  reasoning  is  not  a  logical  process  at  all  ; 
or  it  is  a  formal  inference,  presupposing  the  very  reasoning 
to  which  it  is  prefixed,  and  thus  begging  the  whole  question" 
(Mansel,  Prolegomena  Logica^  p.  217). 

101.  How  far  can  the  legitimacy  of  the  various 
processes  of  Immediate  Inference  be  immediately 
deduced  from  the  laws  of  Identity,  Contradiction 
and  Excluded  Middle } 

Law  of  Identity, — A  is  A, 

I^aw  of  Contradiction, — A  is  not  not-^. 

Law  of  Excluded  Middle, — A  is  either  B  or  not-^. 

We  may  consider  the  application  of  these  laws  to 

(i)     inferences  based  on  the  square  of  opposition ; 

(2)  obversion ; 

(3)  conversion. 

(i)  The  inferences  based  on  the  square  of  opposition 
may  be  considered  to  depend  exclusively  on  the  above 
laws  of  thought.  For  example,  from  the  truth  of  All  S\%  P 
we  may  infer  the  truth  of  Some  *$"  is  /*  by  the  law  of 
Identity,  and  the  falsity  of  some  S  is  not  P  by  the  law  of 
Contradiction ;  from  the  falsity  of  All  S  \%  P  we  may  infer 


CHAP.  VII.] 


PROPOSITIONS. 


the  truth  of  some  5  is  not  P  by  the  law  of  Excluded 
Middle. 

(2)  Obversion  also  may  be  based  entirely  on  the  laws 
of  Contradiction  and  Excluded  Middle.  From  All  S  i^  P 
we  get  No  S  is  not-/*  by  the  law  of  Contradiction ;  and 
from  No  ^  is  P  we  get  All  S  is  not-/*  by  the  law  of  Ex- 
cluded Middle. 

(3)  The  case  of  Conversion  is  different ;  and  I  do  not 
see  how  this  process  can  be  based  exclusively  on  these 
three  laws  of  thought.  Mansel  holds  that  it  can,  but  so 
far  as  I  am  able  to  discover  he  makes  no  attempt  to 
establish  his  position  in  detail.  How,  for  example,  would 
the  application  of  the  three  laws  of  thought  prove  our 
inability  to  convert  an  O  proposition  ?  De  Morgan  appears 
to  me  to  be  perfectly  justified  in  saying, — "When  any 
writer  attempts  to  shew  how  the  perception  of  convertibility 
^A  is  B  gives  BisA*  follows  from  the  principles  of  identity, 
difference  and  excluded  middle,  I  shall  be  able  to  judge  of 
the  process  :  as  it  is,  I  find  that  others  do  not  go  beyond 
the  simple  assertion,  and  that  I  myself  can  detect  the 
pctitio principii  in  every  one  of  my  own  attempts"  {Syllabus,  p. 
47).  The  following  attempt  may  be  taken  as  a  specimen  :  — 
"  All  A  is  ^',  therefore,  some  B  is  A  ;  for  if  no  B  were  A, 
then  A  would  be  both  B  and  not  B,  which  is  impossible." 
It  is  clear  however  that  conversion  is  already  assumed  in 
this  reasoning. 

If  Conversion  cannot  be  based  exclusively  on  the  three 
Laws  of  Thought,  it  follows  that  Contraposition  and  Inver- 
sion cannot  be  based  exclusively  on  these  Laws. 

102.     Proof  of  the  various  rules  of  Conversion. 

The  question  as  to  what  proof  should  be  given  of  the 
various  rules  of  conversion  has  been  partially  discussed  in 
the  two  preceding  sections.    In  them  we  discussed  attempts 

K.  u  8 


114 


PROPOSITIONS. 


[part  II. 


to  prove  conversions  (i)  by  means  of  syllogisms,  (2)  by 
means  of  the  three  laws  of  identity,  contradiction  and  ex- 
cluded middle. 

Bain  writes  as  follows, — "When  we  examine  carefully 
the  various  processes  in  Logic,  we  find  them  to  be  material 
to  the  very  core.  Take  CoJiversion.  How  do  we  know 
that,  if  No  X  is  F,  No  Y  is  JT?  By  examining  cases  in 
detail,  and  finding  the  equivalence  to  be  true.  Obvious  as 
the  inference  seems  on  the  mere  formal  ground,  we  do  not 
content  ourselves  with  the  formal  aspect.  If  we  did,  we 
should  be  as  likely  to  say.  All  ^  is  K  gives  All  Y  is  X-, 
we  are  prevented  from  this  leap  merely  by  the  examination 
of  cases"  {Logic,  Deduction,  p.  251).  The  implication  here 
made  that  the  proof  of  rules  of  conversion  is  a  kind  of 
inductive  proof  seems  to  me  unwarranted. 

The  justification  of  conversion  that  I  should  myself 
give  is  that  in  the  case  of  each  of  the  four  fundamental 
forms  of  proposition,  its  conversion  (or  in  the  case  of  an  O 
proposition,  the  impossibility  of  converting  it)  is  self-evident; 
and  that  we  cannot  go  beyond  this  simple  statement.  Thus, 
taking  an  E  proposition,  I  should  say  that  it  is  self-evident 
that  if  one  class  is  entirely  excluded  from  anotlier  class, 
this  second  class  is  entirely  excluded  from  the  first.  In  the 
case  of  an  A  proposition  it  is  clear  on  reflection  that  the 
statement  All  S  is  P  may  include  one  or  other  of  the  two 
relations  of  classes, — either  S  and  P  coincident,  or  P  con- 
taining kS  and  more  besides,— but  that  these  are  the  only 
two  possible  relations  to  which  it  can  be  applied.  It  is 
self-evident  that  in  each  of  these  cases  some  P  \?>  S)  and 
hence  the  inference  by  conversion  from  an  A  proposition 
is  shewn  to  be  justified'.     In  the  case  of  an  O  proposition, 

^  Compare  section  95,  where  these  inferences  are  illustrated  by  the 
aid  of  the  Eulerian  diagrams. 


CHAP.  VII.] 


PROPOSITIONS. 


115 

if  we  consider  all  the  relationships  of  classes  in  which  it 
holds  good,  we  find  that  nothing  is  true  oi  P  in  terms  of  S 
in  all  of  them.  Hence  O  is  inconvertible'.  I  may  add 
that  I  do  not  see  that  in  the  above  reasoning  we  should 
be  assisted  by  any  explicit  reference  to  the  three  laws  of 
thought;  nor  that  the  application  of  the  three  laws  of 
thought  alone  w^ould  be  sufficient  to  give  us  our  results. 

103.  Without  assuming  Conversion,  how  would  you 
logically  justify  the  process  of  Contraposition  ?     [c] 

^  Again,  compare  section  95. 


8—2 


CHAPTER  VIII. 


1  j» 


PREDICATION   AND    "EXISTENCE  . 


104.  Arc  assumptions  with  regard  to  "  existence  " 
involved  in  any  of  the  processes  of  immediate  infer- 
ence ? 

As  pointed  out  by  Mr  Venn  {Symbolic  Logic,  pp.  127, 
128),  a  discussion  about  ** existence"  need  not  in  this  con- 
nection involve  us  in   any  kind   of  metaphysical   enquiry. 
"  As  to  the  nature  of  this  existence,  or  what  may  really  be 
meant  by  it,  we  have  hardly  any  need  to  trouble  ourselves, 
for   almost   any  possible  sense  in  which  the  logician  can 
understand  it  will  involve  precisely  the  same  difficulties  and 
call  for  the  same  solution  of  them.      We  may  leave  it  to 
any  one  to  define  the  existence  as  he  pleases,  but   when 
he  has  done  this  it  will  always  be  reasonable   to   enquire 
whether  there  is  anything  existing  corresponding  to  the  X 
or  Y  which  constitute  our  subject  and  predicate.     There 
can  in  fact  be  no  fixed  tests  for  this  existence,  for  it  will 
vary  widely  according  to  the  nature  of  the  subject-matter 
with  which  we  are  concerned  in  our  reasonings.     For  in- 
stance, we  may  happen  to  be  speaking  of  ordinary  pheno- 
menal existence,  and  at  the  time  present;  by  the  distinction 

1  It  may  perhaps  be  advisable  for  students,  on  a  first  reading,  to 
omit  this  chapter. 


CHAP.  Vlil.] 


PROPOSITIONS. 


117 


in  question  is  then  meant  nothing  more  and  nothing  deeper 
than  what  is  meant  by  saying  that  there  are  such  things  as 
antelopes  and  elephants  in  existence,  but  not  such  things  as 
unicorns  or  mastodons.  If  again  we  are  referring  to  the 
sum-total  of  all  that  is  conceivable,  whether  real  or  imagi- 
nary, then  we  should  mean  what  is  meant  by  saying  that 
everything  must  be  regarded  as  existent  w^hich  does  not 
involve  a  contradiction  in  terms,  and  nothing  which  does. 
Or  if  we  were  concerned  with  Wonderland  and  its  occu- 
pants we  need  not  go  deeper  down  than  they  do  who  tell 
us  that  March  hares  exist  there.  In  other  words,  the  inter- 
pretation of  the  distinction  wull  vary  very  widely  in  diflferent 
cases,  and  consequently  the  tests  by  which  it  would  have 
in  the  last  resort  to  be  verified ;  but  it  must  always  exist 
as  a  real  distinction,  and  there  is  a  sufficient  identity  of 
sense  and  application  pervading  its  various  significations  to 
enable  us  to  talk  of  it  in  common  terms." 

Now,  several  views  may  be  taken  as  to  what  implication 
with  regard  to  existence,  if  any,  is  involved  in  any  given 
proposition. 

(i)  It  may  be  held  that  every  proposition  implies  the 
existence  of  its  subject,  since  there  is  no  use  in  giving  infor- 
mation with  regard  to  a  non-existent  subject. 

(2)  It  may  be  held  that  although  such  existence  is  gene- 
rally implied,  still  it  is  not  so  necessarily ;  and  that  at  any 
rate  in  Formal  Logic  we  ought  to  leave  entirely  on  one 
side  the  question  of  the  existence  or  the  non-existence  of 
the  subjects  of  our  propositions. 

(3)  The  view  is  taken  by  Mr  Venn  that  for  purposes 
of  Symbolic  Logic,  niiivcrsal  propositions  should  ?iot  be 
regarded  as  implying  the  existence  of  their  subjects,  but 
that /^7;'//r///rtrr  propositions  ^//^///^  be  regarded  as  doing  so. 
This  view  might  be  extended  to  ordinary  Formal  Logic. 


n8 


PROPOSITIONS. 


[part  II. 


Without  at  once  deciding  which  of  these  views  is  to  be 
preferred,  we  may  briefly  investigate  the  consequences  which 
follow  from  them  respectively  so  far  as  immediate  inferences 
are  concerned. 

First,  we  may  take  the  supposition  that  every  proposition 
implies  the  existence  of  its  subject.  Thus,  All  *S  is  /^  implies 
the  existence  of  ..S",  and  it  follows  that  it  also  implies  the 
existence  of  P.  No  S  is  P  implies  the  existence  of  ^S",  and 
since  by  the  law  of  excluded  middle  every  S  is  either  P  or 
not-/*,  it  follows  that  it  also  implies  the  existence  of  not-/*. 

But  now  if  from  All  5  is  /*  we  are  to  be  allowed  to 
obtain  the  ordinary  immediate  inferences, — if,  for  example, 
we  may  infer  All  not-/  is  not-^*, — the  existence  of  not-/*  and 
not  .S*  are  also  involved.  Similarly,  the  conversion  of  No  S 
is  P  requires  that  we  posit  the  existence  of  P  and  not-^". 

On  this  supposition,  then,  we  find  that  propositions  are 
not  amenable  to  the  ordinary  logical  operations^  except  on  the 
assumption  of  the  existence  of  classes  corresponding  not  ?nerely 
to  the  terms  directly  involved  but  also  to  their  contradictories. 

I)e  Morgan  practically  adopts  this  alternative.  "By  the 
ttnii'crse  (of  a  proposition)  is  meant  the  collection  of  all 
objects  which  are  contemplated  as  objects  about  which 
assertion  or  denial  may  take  place.  Let  every  name  which 
belongs  to  the  whole  universe  be  excluded  as  ?ieedless:  this 
must  be  particularly  remembered.  Let  every  object  which 
has  not  the  name  X  {of  ivhich  there  are  ahvays  some)  be 
conceived  as  therefore  marked  with  the  name  x  meaning 
not-AT"  {Syllabus,  pp.  12,  13).  Compare  also  Jevons,  Pure 
Logic,  pp.  64,  65;  Studies  in  Deductive  Logic,  p.  181. 

Secondly,  we  may  take  the  supposition  that  no  proposition 
logically  implies  the  existence  of  its  subject.  On  this  view, 
the  proposition  All  S  is  P  may  be  read,  All  S,  if  there  is 
any  S,  or,  when  there  is  any  S,  is  Pj  and  its  full  implication 


CHAP.  VIII.] 


PROPOSITIONS. 


119 


with  regard  to  existence  may  be  expressed  by  saying  that  it 
denies  the  existence  of  any  thing  that  is  at  the  same  time  S 
and  not  /*.  In  Mr  Venn's  words,  "///<?  burden  of  implication 
of  existence  is  shifted  fro?n  the  affirmative  to  the  negative  form  ; 
that  is,  it  is  not  the  existence  of  the  subject  or  the  predicate 
(in  affirmation)  which  is  implied,  but  the  non-existence  of 
any  subject  which  does  not  possess  the  predicate"  {Symbolic 
Logic,  p.  141).  Similarly,  on  this  view.  No  S  is  /*  implies 
the  existence  neither  of  S  nor  of  /*,  but  merely  denies  the 
existence  of  anything  that  is  both  6"  and  P.  Some  S  is  P 
(or  is  not  /*)  may  be  read  Some  S,  if  there  is  any  S,  is  P 
(or  is  not  P).  Here  we  do  not  even  negative  or  deny  the 
existence  of  any  class  absolutely;  the  sum  total  of  what  we 
affirm  with  regard  to  existence  is  that  if  any  S  exists,  then 
some  P  (or  not-/*)  also  exists. 

Now  having  got  rid  of  the  implication  of  the  existence 
of  the  subject  in  the  case  of  all  propositions,  we  might 
naturally  suppose  that  in  no  case  in  which  we  make  an 
immediate  inference  need  we  trouble  ourselves  with  any 
question  of  "existence"  at  all.  On  further  enquiry,  however, 
we  shall  find  that  so  far  as  particulars  are  obtained,  assump- 
tions with  regard  to  existence  are  still  involved  in  some 
processes  of  immediate  inference. 

All  .S"  is  /*  at  any  rate  implies  that  if  there  is  any  S  there 
is  also  some  /*,  whilst  on  our  present  view  it  does  not  re- 
quire that  if  there  is  any  /*  there  is  also  some  S.  But  the 
converse  of  the  given  proposition, — Some  P  is  S, — does 
imply  this.  "  If  the  predicate  exists  then  also  the  subject 
exists  "  must  therefore  be  regarded  as  an  assumption  which 
is  involved  in  the  conversion  of  an  A  proposition;  similarly, 
in  the  conversion  of  an  I,  and  in  the  contraposition  of  an  E 
or  of  an  O  proposition.  It  follows  also  that  in  passing 
from  All  S  is  P  to  Some  not  .S  is  not-/*,  we  have  to  assume 


I20 


PROPOSITIONS. 


[part  If. 


that  if  there  is  any  not-^*  there  is  also  some  not-P.  It 
does  not  appear  that  there  is  any  similar  assumption  in  the 
conversion  of  an  E  proposition ;  nor  do  I  think  that  there 
is  any  in  the  obversion  either  of  A  or  E,  or  in  the  contra- 
position of  A.  It  might  indeed  at  first  sight  seem  that  in 
passing  from  No  S  is  F  to  All  6*  is  not-/',  we  have  to  assume 
that  if  there  is  any  S  there  is  also  some  not-P.  But,  even 
on  our  present  supposition,  this  is  necessarily  implied  in 
the  proposition  No  S  is  P  itself.  If  there  is  any  S  it  is 
by  the  law  of  excluded  middle  either  P  or  not  P;  therefore, 
given  that  No  S  is  F,  it  follows  immediately  that  if  there 
is  any  S  there  is  some  not-P.  It  can  also  be  shewn  that 
since  No  5  is  P  denies  the  existence  of  anything  that  is 
both  S  and  P,  it  implies  by  itself  that  if  there  is  any  P 
there  is  some  not-^";  and  that  since  the  proposition  All  S 
is  P  denies  the  existence  of  anything  that  is  both  S  and  not- 
/;  it  impHes  by  itself  that  if  there  is  any  not-P  there  is  some 
not-5. 

The  given  supposition  then  provides  for  the  obversion 
and  contraposition  of  A,  and  for  the  obversion  and  con- 
version of  E,  without  any  further  implication  with  regard 
,  to  existence  than  is  contained  in  the  proposidons  them- 
selves. But  the  conversion  or  inversion  of  A  involves 
further  assumptions,  as  shewn  above ;  and  the  same  is  true 
of  the  contraposition  or  inversion  of  E,  the  conversion  of 
I  and  the  contraposition  of  O. 

Now  it  will  be  observed  that  in  the  first  set  of  cases 
we  obtain  by  our  immediate  inference  a  ujiiversal  propo- 
sition ;  in  the  second  set  a  particular  one.  We  may  there- 
fore generalise  our  results  as  follows,— On  the  supposi- 
tion that  no  proposition  logically  implies  the  existence  of  its 
subject  we  do  not  require  to  make  any  assumption  7uith  regard 
to  existence  in  any  process  of  immediate  inferetice  proz'ided  that 


CHAP.  VIII.] 


PROPOSITIONS. 


121 


it  yields  a  universal  conclusion;  but  it  is  generally  other- 
wise in  cases  that  yield  only  a  particular  conclusion.  In 
other  words,  whenever  we  are  left  with  a  universal  con- 
clusion we  need  not  be  afraid  that  any  assumption  with 
regard  to  existence  has  been  introduced  unawares ;  but 
whenever  we  are  left  with  a  particular  conclusion  such  an 
assumption  may  have  been  made,  and  if  we  find  that  it  is 
so,  this  should  be  explicitly  stated. 

T/iirdly,  taking  Mr  Venn's  view,  which  is  the  same  as 
the  preceding  so  far  as  universal  propositions  are  concerned, 
but  which  regards  particular  propositions  as  implying  the 
existence  of  their  subjects,  the  result  just  obtained  hardly 
requires  to  be  modified.  It  must  however  be  observed 
that  on  this  supposition  we  cannot  even  pass  from  All  »S 
is  F  to  Some  S  is  P,  except  under  the  condition  that  the 
existence  of  S  is  granted. 

105.  Shew  that  in  some  processes  of  conversion 
assumptions  as  to  the  existence  of  classes  in  nature 
have  to  be  made;  and  illustrate  by  examining  whether 
any  such  assumptions  are  involved  in  the  inference 
that  if  All  ^  is  P,  therefore  Some  not-^  is  not-P. 

Concerning  this  question,  Professor  Jevons  remarks  that 
it  "must  have  been  asked  under  some  misapprehension. 
The  inferences  of  Formal  Logic  have  nothing  whatever  to 
do  with  real  existence ;  that  is,  occurrence  under  the  con- 
ditions of  time  and  space  "  {Studies  in  Deductive  Logic,  p. 
55).  The  question  is  doubtless  somewhat  unguarded  with 
regard  to  the  nature  of  the  existence  implied,  but  I  think 
that  in  any  case  the  discussion  in  the  preceding  section 
shews  that  it  does  not  admit  of  being  so  summarily  dis- 
missed'.    Even  granting  that  the  formal  logician  may  say 

^  \Vliat  follows  is  to  some  extent  a  repetition  of  what  has  been 


122 


PROPOSITIONS. 


[part  II. 


CHAP.  VIII.] 


PROPOSITIONS. 


123 


that  given  the  proposition  All  S  is  P,  it  is  no  concern  of 
his  whether  or  not  there  are  any  individuals  actually  belong- 
ing to  the  classes  S  and  P,  nevertheless  he  must  admit  that 
the  proposition  at  least  involves  that  if  there  are  any  S  there 
must  be  some  /*,  while  it  does  not  involve  that  if  there  are 
any  P  there  must  be  some  S.  But  now  convert  the  propo- 
sition. We  obtain  Some  P  is  5,  and  this  does  involve  that 
if  there  are  any  P  there  must  be  some  S.  I  do  not  there- 
fore see  how  in  converting  the  given  proposition  this  as- 
sumption can  be  avoided.  Thus,  from  *'All  dragons  are 
serpents",  we  may  infer  by  conversion  '*Some  serpents  are 
dragons,"  and  this  proposition  implies  that  if  there  are 
serpents  there  are  also  dragons.  Similarly,  in  passing  from 
All  S  is  F  to  Some  not-<S  is  not-/^,  it  must  at  least  be 
assumed  that  if  ^S"  does  not  constitute  the  entire  universe 
of  discourse,  neither  does  P  do  so.  If  we  make  immediate 
inferences  from  hypothetical  propositions,  the  necessity  of  a 
similar  assumption  seems  still  more  obvious.  For  example, 
from  the  true  statement  that  if  Governor  Musgrave's  econo- 
mic doctrines  are  correct,  ]\Ir  ^lill  makes  mistakes  in  his 
Political  Economy,  we  can  hardly  without  qualification  infer 
that  in  some  cases  in  which  IVIr  Mill  makes  mistakes  in  his 
Political  Economy,  Governor  Musgrave's  doctrines  are  cor- 
rect, since  Mr  Mill  might  be  sometimes  wrong,  and  never- 
theless Governor  Musgrave  always  so. 

In  another  place  {Studies  in  Deductive  Logic^  p.  141) 
Jevons  remarks,  "  I  do  not  see  how  there  is  in  deductive 
logic  any  ([uestion  about  existence";  and  with  reference  to 
the  opposite  view  taken  by  De  Morgan,  he  says,  "  This  is 
one  of  the  few  points  in  which  it  is  possible  to  suspect  him 

given  in  the  preceding  section.  The  view  that  I  am  here  especially 
combating  however  is  that  Formal  Logic  cannot  possibly  have  any 
concern  with  questions  relating  to  *' existence." 


«; 


r.f 


of  unsoundness."  I  can  however  attach  no  meaning  to 
Jevons's  own  "  Criterion  of  Consistency  "  {Studies  in  Deduc- 
tive Logic^  p.  181)  unless  it  has  some  reference  to  "existence." 
*'  It  is  assumed  as  a  necessary  law  that  every  term  must 
have  its  negative.  This  was  called  the  Laiu  of  Infinity  in 
my  first  logical  essay  {Pure  Logic,  p.  65;  see  also  p.  45);  but 
as  pointed  out  by  Mr  A.  J.  Ellis,  it  is  assumed  by  I)e  Mor- 
gan, in  his  Syllabus,  Article  16.  Thence  arises  what  I  pro- 
pose to  call  the  Criterion  of  Consistency,  stated  as  follows  : — - 
Any  tuio  or  more  propositions  are  contradictory  ic/ien,  and 
only  when,  after  all  possible  substitutions  are  made,  they 
occasion  the  total  disappearance  of  any  term,  positive  or  nega- 
tive, from  the  Logical  Alphabets  What  can  this  mean  but 
that  although  we  may  deny  the  existence  of  the  combination 
AB,  we  cannot  without  contradiction  deny  the  existence  of 
A  itself,  or  not-^,  or  B,  or  not--^?  Indeed,  in  reference 
to  Jevons's  equational  logic  generally,  what  can  negativing  a 
combination  mean  but  denying  its  existence?  For  example, 
I  take  the  following  (juite  at  random, — "There  remain  four 
combinations,  ABC,  aBC,  abC,  and  abc.  But  these  do  not 
stand  on  the  same  logical  footing,  because  if  we  were  to 
remove  ABC,  there  would  be  no  such  thing  as  A  left;  and 
if  we  were  to  remove  abc  there  would  be  no  such  thing  as  c 
left.  Now  it  is  the  Criterion  or  condition  of  logical  con- 
sistency that  every  separate  term  and  its  negative  shall 
remain.  Hence  there  must  exist  some  things  which  are 
described  by  ABC,  and  other  things  described  by  abc''^ 
{Studies  in  Deductive  Logic,  p.  216). 

With  regard  to  Jevons's  criterion  of  consistency  itself,  I 
am  hardly  prepared  to  admit  it.  If  I  am  not  allowed  to 
negative  X,  why  should  I  be  allowed  to  negative  AB'> 
There  is  nothing  to  prevent  X  from  being  itself  a  complex 
term.    In  certain  combinations  indeed  it  may  be  convenient 


124 


TROrOSITIONS. 


[part  II. 


to  substitute  X  for  AB,  or  vice  versa.  It  would  appear  then 
that  what  is  contradictory  when  we  use  a  certain  set  of 
symbols  may  not  be  contradictory  when  we  use  another 
set  of  symbols.  1  should  say  that  Jevons's  criterion  is  some- 
times a  convenient  assumption  to  make,  but  nothing  more 
than  this;  and  it  is  I  think  an  assumption  that  should 
always  be  explicitly  referred  to  when  made. 

106.  Is  a  categorical  proposition  to  be  regarded 
as  logically  implying  the  existence  of  its  subject } 

Our  answer  to  this  question  must  depend  to  some  extent 
on  popular  usage,  and  to  some  extent  on  logical  conveni- 
ence. So  far  as  universal  propositions  are  concerned,  I 
should  be  inclined  on  both  grounds  to  answer  it  in  the 

negative. 

In  the  first  place,  I  do  not  think  that  in  ordinary  speech 
we  always  imi)ly  the  existence  of  the  subjects  of  our  pro- 
positions. No  doubt  we  usually  regard  them  as  existing; 
but  as  Mr  Venn  shews  there  are  undoubtedly  exceptions  to 
this  rule.  "  For  instance,  assertions  about  the  future  do 
not  carry  any  such  positive  presumption  with  them,  though 
the  logician  would  commonly  throw  them  into  precisely  the 
same  '  All  X  is  Y'  type  of  categorical  assertion.  *  Those 
who  pass  this  examination  are  lucky  men '  would  certainly 
be  tacitly  supplemented  by  the  clause  *  if  any  such  there 
be.'  So  too,  in  most  circumstances  of  our  ordinary  life, 
wherever  we  are  clearly  talking  of  an  ideal.  *  Perfectly 
conscientious  men  think  but  little  of  law  and  rule,'  has 
a  sense  without  implying  that  there  are  any  such  men  to 
be  found'  "  {Symbolic  Logic,  pp.  130,  131).    Again,  a  mathe- 

^  The  above  seems  to  me  an  answer  to  such  a  statement  as  the 
following  :— '*  In  an  ordinary  proposition  the  subject  is  necessarily 
admitted  to  exist,  either  in  the  real  or  in  some  imaginary  world  assumed 


CHAP.  VIII.] 


PROPOSITIONS. 


125 


matician  might,  assert  that  a  rectilinear  figure  having  a  mil- 
lion equal  sides  and  inscribable  in  a  circle  has  a  million 
equal  angles,  without  intending  to  imply  the  actual  exist- 
ence of  such  a  figure  ;  or  if  I  know  that  A  h  X,  B  is  Y, 
C  is  Z,  I  may  affirm  that  ABC  is  XYZ  without  wishing 
to  commit  myself  to  the  view  that  the  combination  ABC 
does  ever  really  occur \  Taking  complex  subjects,  and 
limiting  our  conception  of  existence  as  we  not  unfrequently 
do  to  some  particular  universe,  cases  of  this  kind  might  be 
multiplied  indefinitely.  • 

But  if  it  is  granted  that  in  ordinary  thought  the  existence 
of  the  subject  of  the  proposition  sometimes  is  and  some- 
times is  not  implied,  it  follows  that  since  the  logician  cannot 
discriminate  between  these  cases,  he  had  best  content  him- 
self with  leaving  the  question  open,  that  is,  he  should  regard 
such  existence  as  not  necessarily  or  logically  implied. 

And,  further,  to  adopt  this  alternative  is  logically  more 
convenient,  since  so  far  as  the  obtaining  tiniversal  propo- 
sitions by  immediate  inference  is  concerned,  we  do  not  on 
this  supposition  require  any  further  assumptions  with  regard 
to  existence  in  order  that  such  immediate  inference  may  be 
legitimate.     On  the  other  hand,  if  we  take  the  other  alter- 

for  the  nonce When  we  say  No  stone   is   alivc^  or  All  men  are 

mortaly  we  presuppose  the  existence  of  stones  or  of  men.  Nobody 
would  trouble  himself  about  the  possible  properties  of  jiurely  prob- 
lematic men  or  stones"  {Mind,  1876,  pp.  290,  291).  But  the  conclu- 
sions, "  Those  who  pass  this  examination  are  lucky  men,"  "  Perfectly 
conscientious  men  think  but  little  of  law  and  rule"  may  certainly  be 
worth  obtaining,  although  in  the  universe  to  which  reference  is  made, 
(and  in  both  the  cases  in  question  this  would  be  the  actual  material 
universe),  the  subjects  of  these  propositions  might  be  non-existent. 

1  Is  it  not  sometimes  the  case  that  in  order  to  disprove  the  existence 
of  some  combination,  say  AB,  we  establish  a  self-contradictory  pro- 
position of  the  form  AB  is  both  C  and  not-C? 


126 


PROPOSITIONS. 


[part  ii. 


native  and  regard  categorical  propositions  as  always  im- 
plying the  existence  of  their  subjects,  we  have  shewn  in 
section  104  that  we  require  to  assume  the  existence  not 
merely  of  the  actual  terms  involved  in  any  given  proposi- 
tion, but  also  of  their  contradictories. 

The  importance  of  the  question  here  raised  is  more 
particularly  manifest  when  we  arc  dealing  with  very  complex 
propositions,  as  is  shewn  by  IVIr  Venn. 

We  say  then  that  logically  All  S  is  P  implies  only  the 
non-existence  of  anything  that  is  both  S  and  not-P;  No  S 
is  I"  implies  only  the  non-existence  of  anything  that  is  both 

S  and  I\ 

The  case  oi particular  propositions  still  remains;  and 
here  again  I  am  inclined  to  agree  with  the  view  taken  by 
Mr  Venn  in  his  Symbolic  Logic,  namely  that  such  proposi- 
tions should  be  regarded  as  implying  the  existence  of  their 
subjects.  The  chief  grounds  for  adopting  this  view  is  that 
"  an  assertion  confined  to  *  some '  of  a  class  generally  rests 
upon  observation  or  testimony  rather  than  on  reasoning  or 
imagination,  and  therefore  almost  necessarily  postulates 
existent  data,  though  the  nature  of  this  observation  and 
consequent  existence  is,  as  already  remarked,  a  perfectly 
open  question  "  (6>w^rV/V  Z^^i,7V-,  p.  131).  I  doubt  whether 
in  ordinary  speech  we  ever  predicate  anything  of  a  non- 
existent subject  unless  we  do  so  universally.  The  principal 
objection  to  this  view  is  perhaps  the  paradox  which  follows 
from  it,  namely  that  we  are  not  without  qualification  justified 
in  inferring  from  All  S  \s  P  that  Some  S  is  /*,  (since  the 
latter  proposition  implies  the  existence  of  *S',  while  the 
former  does  not).  It  may  even  be  said  tliat  this  view 
practically  banishes  the  particular  proposition  from  Logic 
altogether.  Possibly  if  it  were  so,  it  would  be  no  very 
serious  matter.     But  I  do  not  think  that  it  is  so.     We  have 


CHAP.  VIII.] 


PROPOSITIONS. 


127 


only  to  be  careful  in  using  such  propositions  to  note  the 
assumption  involved  in  their  use.  The  principal  value  of 
particulars  is  in  their  relation  of  contradiction  to  universals 
of  different  quality.  But  their  use  in  this  respect  is  entirely 
consistent  with  the  above.  We  have  taken  the  view  that 
the  import  of  All  ^  is  /^  is  to  deny  that  there  is  any  S  that 
is  not-/*;  we  are  now  taking  the  view  that  the  import  of 
Some  5  is  not  P  is  to  affirm  that  there  is  some  S  that  is 
not-/*.  This  clearly  brings  out  the  contradictory  character 
of  the  two  propositions.     Similarly  with  I  and  E. 

One  interesting  point  to  notice  here  is  that  if  there  is  no 
implication  of  the  existence  of  the  subject  in  universal  pro- 
positions we  are  not  actually  precluded  from  asserting  to- 
gether two  contraries.  We  may  say  All  S\s>  P  and  No  S  is 
P'j  but  this  virtually  is  to  deny  the  existence  of  ^S*. 

All  S  is  P  excludes 


No  S  is  P  excludes 

0, 

But  these  are  all  possible  cases. 

In  other  respects,  this  investigation  if  pursued  might 
somewhat  modify  accepted  logical  doctrines  ;  but  I  feel 
convinced  that  we  should  be  ultimately  left  with  a  consistent 
whole. 

The  truth  is,  as  Mr  Venn  has  remarked,  that  most 
English  logicians  have  made  no  critical  examination  at  all 


128 


PROPOSITIONS. 


[part  II. 


CHAP.  VIII.] 


PROPOSITIONS. 


129 


of  the  question  here  raised.  It  may  be  desirable  to  return 
to  it  briefly  in  connection  with  the  syllogism.  Compare 
sections  273 — 277. 

[The  above  view,  which  is  taken  by  Mr  Venn  in  respect 
to  Symbolic  Logic,  and  which  I  have  attempted  to  apply 
to  ordinary  Formal  Logic,  is  practically  identical  with  that 
somewhat  recently  put  forward  in  a  more  paradoxical  form 
by  Professor  Brentano.  Compare  Alind^  1876,  pp.  289 — 292. 
*' Where  we  say  Some  man  is  sick,  Brentano  gives  as  a  sub- 
stitute, T/ierc  is  a  sick  man.  Instead  of  No  stone  is  alive, 
he  puts  There  is  no/  a  live  stone.  Some  man  is  not  learned 
becomes  There  is  an  unlearned  man.  Finally,  All  men  are 
mortal  is  to  be  expressed  in  his  system  There  is  not  an  im- 
mortal ;;/j;/."] 

107.  Discuss  the  relation  between  the  propositions 
All  5  is  P  and  All  not-5  is  1\ 

This  is  an  interesting  case  to  notice  in  connection  with 
the  discussion  raised  in  the  preceding  sections. 

All  S  is  P=  No  ^  is  not-/'-  No  not-P  is  5. 

All  not-^Sis  /"^No  not-^"  is  not-/'- No  not-Pis  not-5 

=  All  not-P  is  S. 

The  given  propositions  come  out  therefore  as  contraries. 

(i)  On  the  view  that  we  ought  not  to  enter  into  any 
discussion  concerning  "existence"  in  connection  with  im- 
mediate inference,  we  must  I  suppose  rest  content  with 
this  statement  of  the  case.  It  seems  however  sufficiently 
curious  to  demand  further  investigation  and  explanation. 

(2)  On  the  view  that  propositions  imply  the  existence 
of  their  subjects,  we  have  shewn  in  section  104,  that  we 
are  not  justified  in  passing  from  All  not-^"  is  P  to  All  \\o\.-P 
is  S  unless  we  assume  the  existence  of  not-P.     But  it  will 


be  observed  that  in  the  case  before  us,  the  given  propo- 
sitions make  such  an  assumption  unjustifiable.  Since  All  *S 
is  P  and  All  not- 5  is  P,  and  everything  is  either  S  or  not-S 
by  the  law  of  excluded  middle,  it  follows  that  nothing  is 
not-/*. 

In  reducing  the  given  propositions  therefore  to  such  a 
form  that  they  appear  as  contraries,  (and  therefore  as  in- 
consistent with  each  other),  we  assume  the  very  thing  that 
taken  together  they  really  deny. 

(3)  On  the  view  that  at  any  rate  universal  propositions 
do  not  imply  the  existence  of  their  subjects,  we  have  shewn 
in  the  preceding  section,  that  the  propositions  No  not-/*  is 
Sy  All  not-/  is  S,  are  either  inconsistent  or  else  they  express 
the  fact  that  P  constitutes  the  entire  universe  of  discourse. 
But  this  fact  is  the  very  thing  that  is  given  us  by  the  propo- 
sitions in  their  original  form. 

On  either  of  the  views  (2)  or  (3),  then,  the  result  obtained 
is  satisfactorily  accounted  for  and  explained. 


K.  U 


CHAP.  IX.] 


PROPOSITIONS. 


131 


CHAPTER  IX. 

HYrOTHETICAL    AND    DISJUNCTIVE    PROPOSITIONS. 

108.  The  nature  of  the  logical  distinction  between 
Categorical  and  Hypothetical  Propositions. 

Are  the  propositions  "All  B  is  C  and  "If  any- 
thing is  B,  it  is  (7"  equivalent.^  or  can  either  be 
inferred  from  the  other  1 

Mr  Venn  holds  that  the  real  diffcraitta  of  Hypothetical 
Propositions  is  **to  express  human  doubt"  {Mifuf,  1879, 
p.  42).  I  should  myself  prefer  to  express  the  import  of 
Hypothetical  Propositions  by  saying  that  they  affirm  a 
connection  between  certain  events,  whenever  they  happen 
or  if  they  ever  happen,  whilst  leaving  the  question  en- 
tirely open  whether  or  not  they  do  ever  happen.  The 
doubt  which  they  imply  is  rather  incidental,  than  the 
fundamental  or  differentiating  characteristic  belonging  to 
them.  Materially  indeed  I  think  that  they  do  sometimes 
imply  the  actual  occurrence  of  their  antecedents.  When- 
ever the  connection  between  the  antecedent  and  the  con- 
sequent in  a  hypothetical  proposition  can  be  inferred 
from  the  nature  of  the  antecedent  independently  of  specific 
experience,  (and  this  may  be  the  more  usual  case),  then  the 
actual  happening  of  the  antecedent  is  not  in  any  sense  in- 


volved; but  if  our  knowledge  of  the  connection  does  depend 
on  specific  experience,  (as  it  sometimes  may),  and  could  not 
have  been  otherwise  obtained,  then  such  actual  happening 
would  appear  to  be  materially  involved.  For  example,  the 
statement,  '*  If  we  descend  into  the  earth,  the  temperature 
increases  at  a  nearly  uniform  rate  of  i*'  Fahr.  for  every 
50  feet  of  descent  down  to  almost  a  mile,"  requires  that 
actual  descents  into  the  earth  should  have  been  made,  for 
otherwise  the  truth  of  the  statement  could  not  have  been 
known. 

It  may,  however,  be  replied  that  the  doubt  applies  to  the 
actual   occurrence  of  the  antecedent   m  a  given   i?ista?tce. 
When  I  say  "  If  the  glass  falls,  it  will  rain,"  I  imply  doubt  as 
to  whether  it  actually  will  fall  on  the  occasion  to  which  I  am 
'r^'  referring.     (Compare  Venn,  Symbolic  Logic,  pp.  331 — ZZZ-) 

But  may  not  this  be  the  case  also  with  categorical  propo- 
sitions? For  example,  if  I  am  in  doubt  w^hcther  a  given 
plant  Is  an  orchid,  I  may  apply  the  proposition  "All  orchids 
have  opposite  leaves"  in  order  to  resolve  my  doubt.  We 
have  such  a  case  as  this  whenever  categorical  propositions 
are  used  in  the  process  of  diagnosis,  and  it  can  hardly  be 
said  that  we  never  do  employ  categorical  propositions  in  this 
manner. 

Still,  it  is  clear  that  the  hypothetical  proposition  does  not 
necessarily  imply  the  actual  occurrence  of  its  antecedent; 
and  therefore,  if  the  view  is  taken  that  the  categorical  pro- 
position does  necessarily  imply  the  actual  existence  of  its 
subject,  (compare  sections  104,  106),  w^e  have  a  marked 
distinction  between  the  two  kinds  of  propositions.  "If 
anything  is  B,  it  is  C"  cannot  be  resolved  into  "All  B  is 
C",  since  the  latter  implies  the  existence  of  B  while  the 
former  does  not. 

Another  view  with  regard  to  categorical  propositions, 

9-2 


132 


PROPOSITIONS. 


[part  II. 


and  the  one  for  which  I  have  expressed  a  preference,  is  that 
they  do  not  necessarily  imply,  (and  therefore  do  not  logically 
imply),  the  existence  of  their  subjects.  On  this  view,  I  do 
not  see  that  we  have  any  logical  distinction  between  hypo- 
thetical and  categorical  propositions,  except  a  distinction  of 
form  ;  that  is,  they  may  be  resolved  into  one  another.  We 
may  say  indifferently  *'A11  B  is  C"  or  "If  anything  is  ^dt 
is  C"  j  "  If  AisB,  C  is  Z>"  or  '*  All  cases  of  A  being  B  are 
cases  of  C  being  Z>." 

Kant  denies  that  we  can  reduce  the  hypothetical  judg- 
ment to  the  categorical  form  on  the  following  ground  :  *'  In 
categorical  judgments  nothing  is  problematical,  but  every- 
thing assertative;  in  hypothetical  it  is  merely  the  connection 
between  the  antecedent  and  the  consequent  that  is  assertative. 
Hence  here  we  may  combine  two  false  judgments."     This 
view  has  I  think  been  virtually  discussed  in  what  I  have 
already  said.     If  the  categorical  judgment  is  regarded  as 
affirming  not  merely  a  connection  between  the  subject  and 
the  predicate  but  also  the  existence  of  the  subject,  then  I 
admit  the  force  of  the  above  argument,  and  allow  that  the 
hypothetical  judgment  cannot  be  reduced  to  the  categorical 
form.     But  //  /lie  categorical  judgment  is  not  regarded  as 
affirming  the  existence  of  the  subject,  it  (like  the  hypothetical 
judgment)  asserts  no  more  than  a  connection  ;  it  is  no  more 
assertative   than   the   hypothetical  judgment,  and  just   as 
problematic.     The  non-existence  of  the  subject  of  the  cate- 
gorical corresponds  exactly  to  the  falsity  of  the  antecedent 
of  the  hypothetical ;  and  if  in  the  latter  we  may  combine 
two  false  judgments,  in  the  former  we  may  combine  two 
non-existent  entities.    I  may  say,  If  A  is  B,  C  is  Z>,  although 
A  is  B  is  a  false  judgment ;  but  similarly  I  may  say  any 
case   of  A   being  ^   is   a  case  of   C  being  Z>,  although 
the  case  of  A  being  B  is  3l  non-existent  case.     I  cannot 


"/A 


CHAP.  IX.] 


PROPOSITIONS. 


133 


see  that  in  the  latter  of  these  statements  I  have  committed 
myself  to  anything  whatever  that  is  not  contained  in  the 
former. 

Hamilton  also  (Logic,  i.  p.  239)  holds  that  a  hypothetical 
judgment  cannot  be  converted  into  a  categorical.  "The 
thought,  A  is  through  B,  is  wholly  different  from  the  thought, 
A  is  in  B.  The  judgment, — If  God  is  righteous,  then  will 
the  wicked  be  punished,  and  the  judgment, — A  righteous 
God  punishes  the  wicked,  are  very  different,  although  the 
matter  of  thought  is  the  same.  In  the  former  judgment, 
the  punishment  of  the  wicked  is  viewed  as  a  consequent 
of  the  righteousness  of  God ;  whereas  the  latter  considers 
it  as  an  attribute  of  a  righteous  God.  But  as  the  conse- 
quent is  regarded  as  something  dependent  from, — the  at- 
tribute, on  the  contrary,  as  something  inhering  in,  it  is 
from  two  wholly  different  points  of  view  that  the  two 
judgments  are  formed."  Now  it  must  certainly  be  admitted 
that  in  any  given  instance  there  are  reasons  why  we  choose 
the  hypothetical  mode  of  expression  rather  than  the  cate- 
gorical, or  vice  versa ;  but  the  only  question  that  concerns 
us  from  a  logical  point  of  view  is  whether  precisely  the 
same  meaning  cannot  be  expressed  in  either  form.  Plamilton 
would  appear  to  deny  not  merely  that  a  hypothetical 
judgment  can  be  converted  into  a  categorical,  but  also  that 
a  categorical  can  be  converted  into  a  hypothetical.  But, 
(leaving  on  one  side  the  question  of  the  existence  of  the 
subject  in  a  categorical  proposition,  which  has  already  been 
discussed),  can  any  one  who  allows  that  "all  orchids  have 
opposite  leaves"  deny  that  "if  this  plant  is  an  orchid  it 
has  opposite  leaves"?  Can  any  one  who  allows  that  "if 
there  are  sharpers  in  the  company  we  ought  not  to  gamble," 
deny  that  "all  cases  in  which  there  are  sharpers  in  the 
company  are  cases  in  vrhich  we  ought  not  to  gamble"? 


134 


PROPOSITIONS. 


[part  II. 


If  this  is  admitted,  the  logical  question  is  to  my  mind  dis- 
posed of. 

No  doubt  hypothetical  propositions  will  frequently  look 
awkward  when  expressed  in  the  categorical  form,  but  in 
some  cases  logical  error  is  more  likely  to  be  avoided  if  we 
reduce  them  to  this  form  before  manipulating  them;  and 
I  cannot  see  how  we  lose  anything,  or,  (on  the  view  now 
taken  with  regard  to  the  existential  import  of  categorical 
propositions),  imply  anything  that  we  should  not  imply,  in 
so  dealing  with  them.  I  have  given  examples  shewing  that 
the  doctrines  of  opposition  and  immediate  inference  may  be 
applied  to  hypothetical.  We  shall  find  that  the  same  is 
true  of  the  doctrine  of  syllogism,  though  it  may  be  useful 
to  frame  special  rules  when  we  are  dealing  with  propositions 
expressed  in  this  form. 

109.  The  interpretation  of  Disjunctive  Proposi- 
tions. 

There  is  a  difference  of  opinion  among  logicians  as  to 

^  Mansel's  view  upon  this  question  [AUrich,  pp.  103,  104)  is  not 
easy  to  understand.  lie  admits  however  that  "  If  ^f  is  By  C  \s  Z> " 
implies  that  "  Every  case  of  A  being  ^  is  a  case  of  C  being  Z>."  lie 
even  goes  so  far  as  to  resolve  '*  If  all  A  is  B,  all  A  is  C"  into  "All 
B  is  C,"  which  is  clearly  erroneous.  His  whole  treatment  of  hypo- 
theticals  is  puzzling.  For  example,  he  says,  "The  judgment,  *  K  A 
is  Bf  C  is  Z>,'  asserts  the  existence  of  a  consequence  necessitated  by 
laws  other  than  those  of  thought,  and  consec[uently  out  of  the  province 
of  Logic"  (A/Jn'i/iy  p.  236;  Prolegomena  Logiea,  p.  230).  But 
similarly  a  categorical  proposition  may  assert  a  connection  not  neces- 
sitated by  laws  of  thought ;  and  I  do  not  see  that  we  have  here  any 
reason  for  subjecting  hypothetical  propositions  to  a  peculiar  treatment. 
I  am  inclined  to  think  that  what  makes  Mansel's  discussion  of  hypo- 
thetical propositions  so  difficult  is  that  he  attempts  to  apply  to  them 
the  strict  conccjHualist  view  of  Logic,  which  it  is  impossible  to  apply 
consistently  throughout  without  divesting  Logic  of  all  content  what- 
soever. 


CHAP.  IX.] 


PROPOSITIONS. 


135 


whether  the  alternatives  in  a  disjunctive  proposition  should 
be  regarded  as  mutually  exclusive.  For  example,  in  the 
proposition  A  is  either  B  or  C,  there  is  not  general  agree- 
ment as  to  whether  it  is  logically  implied  that  A  cannot  be 
both  B  and  C  \ 

There  are  at  least  two  questions  involved  which  should 
be  distinguished. 

(i)  In  ordinary  speech  do  we  intend  that  the  alter- 
natives in  a  disjunctive  proposition  should  be  necessarily 
understood  as  excluding  one  another.'*  A  very  few  instances 
will  I  think  enable  us  to  answer  this  question  in  the 
negative.  *'Take,  for  instance,  the  proposition — 'A  peer 
is  either  a  duke,  or  a  marquis,  or  an  earl,  or  a  viscount,  or 
a  baron '...Yet  many  peers  do  possess  two  or  more  titles, 
and  the  Prince  of  Wales  is  Duke  of  Cornwall,  Earl  of 
Chester,  Baron  Renfrew,  &c....In  the  sentence — *  Repent- 
ance is  not  a  single  act,  but  a  habit  or  virtue,'  it  cannot  be 
implied  that  a  virtue  is  not  a  habit... Milton  has  the  ex- 
pression in  one  of  his  Sonnets — '  Unstain'd  by  gold  or  fee,' 
where  it  is  obvious  that  if  the  fee  is  not  always  gold,  the 
gold  is  a  fee  or  bribe.  Tennyson  has  the  expression 
*  wreath  or  anadem.'  Most  readers  would  be  quite  un- 
certain whether  a  wreath  may  be  an  anadem,  or  an  anadem 
a  wreath,  or  whether  they  are  quite  distinct  or  quite  the 
same"  (Jevons,  Pure  Logic,  pp.  76,  77). 

(2)  But  this  does  not  absolutely  settle  the  question. 
It  may  be  said: — Granted  that  in  common  speech  the 
alternatives  of  a  disjunction  may  or  may  not  be  mutually 
exclusive,  still  in  Logic  we  should  be  more  precise,  and 


'  Whatcly,  Mansel,  Mill,  and  Jevons  would  answer  this  question 
in  the  negative ;  Kant,  Hamilton,  Thomson,  Boole,  Bain,  and  Fowler 
in  the  affirmative. 


136  PROPOSITIONS.  [part  ii. 

the  statement  "^  is  either  B  or  C"  (where  it  may  be  both) 
should  be  written  ^'  A  is  either  B  ox  C  ox  both." 

This  is  a  question  of  interpretation  or  method,  and  I  do 
not  apprehend  that  any  burning  principle  is  involved  in  the 
answer  that  we  may  give.  For  my  own  part  I  do  not 
find  any  reason  for  diverging  from  the  usage  of  everyday 
language.  On  the  other  hand,  I  think  that  if  Logic  is 
to  be  of  practical  utility,  the  less  logical  forms  diverge 
from  those  of  ordinary  speech  the  better.  And  further,  it 
conduces  to  clearness  if  we  make  a  logical  proposition 
express  as  little  as  possible.  "  A  is  either  B  or  C,  it  can- 
not be  both"  is  best  given  as  two  distinct  propositions'. 

^  A  view  strongly  opposed  to  that  adopted  in  the  text  is  taken  in  a 
recently  published  work  on  the  Principles  of  Logic  by  Mr  Bradley  of 
Merton  College,  Oxford.  Ilis  argument  is  as  follows: — "The  com- 
monest way  of  regarding  disjunction  is  to  take  it  as  a  combination  of 
hypotheses.  This  view  in  itself  is  somewhat  superficial,  and  it  is 
possible  even  to  state  it  incorrectly.  '  Either  A  \s  B  ox  C  \s  Z>'  means, 
we  are  told,  that  if  A  is  not  B  then  C  is  Z>,  and  if  C  is  not  D  then 
A  is  B.  But  a  moment's  reflection  shews  us  that  here  two  cases  are 
omitted.  Supposing,  in  the  one  case,  that  A  is  B,  and  supposing, 
in  the  other,  that  C  is  Z>,  are  we  able  in  these  cases  to  say  nothing  at 
all?  Our  'either — or'  can  certainly  assure  us  that,  HA  is  By  C—D 
must  be  false,  and  that,  if  C  is  D,  then  A — B  is  false.  We  have 
not  exhausted  the  disjunctive  statement,  until  we  have  provided  for 
four  possibilities,  B  and  noi-B,  C  and  not-C"  {Principles  of  Logic^ 
p.  I2i).  The  question  raised  is  really  one  of  interpretation,  as  I  have 
indicated  above;  but  this  is  what  Mr  Bradley  will  not  admit.  In  my 
view,  it  is  open  to  a  logician  to  choose  either  of  the  two  ways  of 
interpreting  a  disjunctive  proposition,  provided  that  he  makes  it  quite 
clear  which  he  has  selected  ;  but  I  can  see  no  good  in  dogmatising  as 
in  the  following  passage, — "Our  slovenly  habits  of  expression  and 
thought  are  no  real  evidence  against  the  exclusive  character  of  dis- 
junction. M  is  ^  or  c^  does  strictly  exclude  M  is  both  b  and  r.' 
When  a  speaker  asserts  that  a  given  person  is  a  fool  or  a  rogue,  he 
may  not  mean  to  deny  that  he  is  both.     But,  having  no  interest  in 


CHAP.  IX.] 


PROPOSITIONS. 


137 


I 


Professor  Fowler  indicates  this  view  in  his  statement  that 
*'  it  is  the  object  of  Logic  not  to  state  our  thoughts  in 
a  condensed  form  but  to  analyse  them  into  their  simplest 
elements"  {Deductive  Logic,  p.  32);  though  he  does  not 
apply  it  to  the  case  before  us. 

Mansel  arguing  in  favour  of  the  view  that  I  have  taken 
remarks, — "But  let  us  grant  for  a  moment  the  opposite 
view,  and  allow  that  the  proposition,  *  All  C  is  either  A  or 
B^  implies,  as  a  condition  of  its  truth,  '  No  C  can  be  both.' 
Thus  viewed,  it  is  in  reality  a  complex  proposition,  contain- 
ing two  distinct  assertions,  each  of  which  may  be  the  ground 
of  two  distinct  processes  of  reasoning,  governed  by  two 
opposite  laws.  Surely  it  is  essential  to  all  clear  thinking, 
that  the  two  should  be  separated  from  each  other,  and  not 
confounded  under  one  form  by  assuming  the  Law  of  Ex- 
cluded Middle  to  be,  what  it  is  not,  a  complex  of  those 
ofldentity  and  Contradiction"  {Prolegomena  Logica,^.  238). 

Of  course  if  the  alternatives  are  logical  contradictories 
they  are  logically  exclusive,  but  otherwise  in  the  treatment 
of  disjunctive  propositions  in  the  following  pages  I  do  not 
regard  diem  as  being  so.  If  in  any  case  they  happen  to  be 
materially  incompatible,  this  must  be  separately  stated. 

110.  From  the  statement  that  blood-vessels  are 
either  veins  or  arteries,  does  it  follow  logically  that  a 
blood-vessel,  if  it  be  a  vein,  is  not  an  artery  t  Give 
your  reasons.  [i-] 


shewing  that  he  is  both,  being  perfectly  satisfied  provided  he  is  one, 
either  b  or  r,  the  speaker  has  not  the  possibility  be  in  his  mind.  Ig- 
noring it  as  irrelevant,  he  argues  as  if  it  did  not  exist.  And  thus  he 
may  practically  be  right  in  what  he  says,  though  formally  his  statement 
is  downright  false :  for  he  has  excluded  the  alternative  be''  (p.  124). 


138 


PROPOSITIONS. 


[part  II. 


CKAP.  IX.] 


PROPOSITIONS. 


139 


111.  Put,  if  you  can,  the  whole  meaning  of  a  dis- 
junctive proposition  (such  as,  Iiither  A  \s  B  or  C  \s  D) 
in  the  form  of  a  single  and  simple  Hypothetical,  and 
prove  your  expression  to  be  sufficient.  [r.] 

Adopting  the  view  that  in  a  disjunctive  proposition  the 
alternatives  are  not  to  be  regarded  as  necessarily  excluding 
one  another,  such  a  disjunctive  proposition  as  the  above  is 
primarily  reducible  to  two  hypotheticals,  namely,  \i  A  is  not 
B,  C  is  Z>,  and  If  C  is  not  D,  A  is  B.  But  each  of  these 
is  the  contrapositive  of  the  other,  and  may  therefore  be  in- 
ferred from  it.  Hence  the  full  meaning  of  the  disjunctive 
is  expressed  by  means  oi  either  of  these  hypotheticals  ^ 

Professor  Croom  Robertson  called  attention  to  this 
point  in  Alind,  1877,  p.  266, — "The  other  form  of  propo- 
sition ranged  by  logicians  with  the  Hypothetical,  namely  the 
Disjunctive,  may  be  shewn  to  be  as  simple  as  the  pure 
Hypothetical  being  in  fact  a  special  case  of  it.  The  com- 
mon view  is  that  it  involves  at  least  two  hypothetical  propo- 
sitions, or,  as  some  say,  even  four.  Thus  *  Either  A  is  B 
or  C  is  Z> '  is  resolved  by  some  into  the  four  hypotheticals — 

^  Mr  Eradley  {Priuciples  of  Loc^ic^  p.  121),  lays  it  clown  that 
*'  disjunctive  judgments  cannot  really  be  reduced  to  hypotheticals " 
at  all;  but  I  hardly  care  to  disagree  with  him  since  he  admits  all  that 
I  should  contend  for.  He  distinctly  resolves  **//  is  b  or  <:"  into 
hypotheticals  (p.  130);  but,  he  adds,  although  the  meaning  of  dis- 
junctives can  thus  ''be  given  hypothetically ;  we  must  not  go  on  to 
argue  from  this  that  they  ^r^  hypothetical"  (p.  121).  They  ''declare 
a  fact  without  any  supposition  "  (p.  122).  But  so  does  the  hypothetical 
itself,  namely,  the  connection  between  the  antecedent  and  the  conse- 
quent. Further,  "A  combination  of  hypotheticals  surely  does  not  lie 
in  the  hypotheticals  themselves"  (p.  122).  Undoubtedly,  by  means  of 
a  combination  of  hypotheticals,  we  may  make  a  most  categorical  state- 
ment ;  e.g,,  If  A  is  B,  CisD;  and  if  A  is  not  B,  C'ls  D, 


\ 


KAisB,  CisnotZ>(i), 

If  A  is  not  B,  C  is  £>  (2), 

If  Cis  D,  A  is  not  B  (3), 

If  C  is  not  Bf,  AisB  (4), 

— but  the  first  and  third  of  these  are  rejected  by  others,  and 
with  reason,  because  they  are  in  fact  implied  only  when  the 
alternatives  are  logical  opposites.  The  remaining  propo- 
sitions (2)  and  (4)  are,  however,  the  logical  contrapositivcs 
of  one  another;  and  this  amounts  to  saying  that  either  of 
them  /^  I'/se//  is  a  full  and  adequate  expression  of  the 
original  disjunctive." 


CHAP.  I.] 


SYLLOGISMS. 


141 


PART     III. 

SYLLOGISMS. 


CHAPTER    L 

THE   RULES   OF   THE   SYLLOGISM. 

112.     The  Terms  of  the  Syllogism. 

A  reasoning  consisting  of  three  categorical  propositions 
(of  which  one  is  the  conchision),  and  containing  three  and 
only  three  terms,  is  called  a  Categorical  Syllogism. 

Every  categorical  syllogism  then  contains  three  and  only 
three  terms,  of  which  two  appear  in  the  conclusion  and  also 
in  one  or  other  of  the  premisses,  and  one  in  the  premisses 
only.  That  which  appears  as  the  i)redicate  of  the  conclusion, 
and  in  one  of  the  premisses,  is  called  the  inajor  term;  that 
which  appears  as  the  subject  of  the  conclusion,  and  in  one 
of  the  premisses,  is  called  the  minor  term;  and  that  which 
appears  in  both  the  premisses,  but  not  in  the  conclusion, 
(being  that  term  by  their  relations  to  which  the  mutual 
relation  of  the  two  other  terms  is  determined),  is  called  the 
middle  term. 


r 


Thus,  in  the  syllogism, — 

All  M  is  P, 

All  S  is  J/, 

therefore,  All  Sh  P\ 

P  is  the  major  term,  S  is  the  minor  term,  and  M  is  the 
middle  term. 

[These  respective  designations  of  the  terms  of  a  syllogism 
resulted  from  such  a  syllogism  as, — 

All  M  is  P, 

All  S  is  M, 

therefore.  All  S  is  P, 

being  taken  as  the  type  of  syllogism.  With  the  exception 
of  the  somewhat  rare  case  in  which  the  terms  of  a  propo- 
sition are  coextensive,  such  a  syllogism  as  the  above  may  be 
represented  by  the  following   diagram.      Here   clearly  the 


major  term  is  the  largest  in  extent,  and  the  minor  the 
smallest,  while  the  middle  occupies  an  intermediate  position. 
But  we  have  no  guarantee  that  the  same  relation  between 
the  terms  of  a  syllogism  will  hold,  when  one  of  the  pre- 
misses is  a  negative  or  a  particular  proposition;  e.g.^  the 
following  syllogism, — 

No  M  is  P, 

All  S  is  M, 

therefore,  No  6"  is  /*, 


142 


gives  as  one  case 


SYLLOGISMS. 


[part  III. 


where  the  major  term  may  be  the  smallest  in  extent,  and  the 
middle  the  largest. 

Again,  the  following  syllogism, — 

No  M  is  P, 
Some  S  is  M, 
therefore,  Some  S  is  not  P^ 
gives  as  one  case 


where  the  major  term  may  be  the  smallest  in  extent  and  the 
minor  the  largest. 

AVith  regard  to  the  middle  term,  however,  we  may  note 
that  although  it  is  not  always  a  middle  term  in  extent,  it  is 
always  a  middle  term  in  the  sense  that  by  its  means  the  two 
other  terms  are  connected,  and  their  mutual  relation  deter- 
mined.] 

113.     The  Propositions  of  the  Syllogism. 

Every  categorical  syllogism  consists  of  three  propositions. 
Of  these  one  is  the  conclusion.  The  premisses  are  called 
the  major  premiss  and  the  minor  premiss  according  as  they 
contain  the  major  term  or  the  minor  term  respectively. 


CHAP.  I.] 


SYLLOGISMS. 


143 


Thus,  All  M  is  P,  (major  premiss), 
All  S  is  J/,  (minor  premiss), 
therefore.  All  S  is  P,  (conclusion). 

It  is  usual,  (as   in  the  above  syllogism),  to   state   the 
major  premiss  first  and  the  conclusion  last. 

114.     The  Rules  of  the  Syllogism;  and  the  Deduc- 
tion of  the  Corollaries. 

The   rules  of  the  Syllogism  as  usually  stated   are   as 
follows  :^ 

(i)     Every  syllogism  contains  three  and  only  three  terms. 

(2)  Every  syllogism  consists  of  three  and  only  three  pro- 
positions. 

It  may  be  observed  that  these  are  not  so  much  rules,  as 
a  general  description  of  the  nature  of  the  syllogism.  A 
reasoning  which  does  not  fulfil  these  conditions  may  be 
formally  valid,  but  we  should  not  call  it  a  syllogism'.  The 
four  following  rules  are  really  rules  in  the  sense  that  if, 
when  we  have  got  the  reasoning  into  the  form  of  a  syl- 
logism, they  are  not  fulfilled,  then  the  reasoning  is  invalid. 

(3)  No  one  of  the  three  terms  of  the  syllogism  must  be 
used  ambiguously ;  and  the  middle  term  must  be  distributed 
once  at  least  in  the  premisses. 

This  rule  is  frequently  given  in  the  form:  "The  middle 
term  must  be  distributed  once  at  least,  and  must  not  be 
ambiguous,"  {e.g.,  in  Jevons,  Elementary  Lessons,  p.  127). 

^  For  example,  B  is  greater  tlian  C, 

A  is  greater  than  B^ 

tliereforc,  A  is  greater  than  C. 

Here  there  are  four  terms,  since  the  predicate  of  the  second  premiss 
is  "greater  than  ^,"  and  this  is  not  the  same  as  the  subject  of  the 
first  premiss  "^." 


144 


SYLLOGISMS. 


[part  III. 


But  it  is  obvious  that  we  must  guard  against  ambiguous 
major  and  ambiguous  minor  as  well  as  against  ambiguous 
middle. 

If  the  middle  term  is  distributed  in  neither  of  the  pre- 
misses, the  syllogism  is  said  to  be  subject  to  the  fallacy  of 
undistributed  in  id  die. 

(4)  No  term  must  he  distributed  in  the  conclusion  which 
was  7iot  distributed  in  o?ie  of  the  premisses. 

The  breach  of  this  rule  is  called  illicit  process  of  the 
major,  or  illicit  process  of  the  minor,  as  the  case  may  be;  or, 
more  briefly,  illicit  major  or  illicit  minor, 

(5)  From  tivo  negative  premisses  nothing  can  be  inferred. 

(6)  If  one  premiss  is  negative,  the  conclusion  must  be  fiega- 
tive;  and  to  prove  a  negative  conclusion,  one  of  the  premisses 
must  be  negative. 

From  these  rules,  three  corollaries  may  be  deduced : — 

(i)  From  two  particular  premisses  nothing  can  be  in- 
ferred. 

Two  particular  premisses  must  be  either 
(a)    both  negative, 
or         (fi)  both  affirmative, 
or         (7)  one  negative  and  one  affirmative. 

But  in  case  (a),  no  conclusion  follows  by  rule  5. 

In  case  (/?),  since  no  term  can  be  distributed  in  two 
particular  affirmative  propositions,  the  middle  term  cannot 
be  distributed,  and  therefore  no  conclusion  follows  by  rule  3. 

In  case  (y),  if  we  can  have  a  conclusion  it  must  be  nega- 
tive (rule  6).  The  major  term  therefore  will  be  distributed 
in  the  conclusion  ;  and  hence  we  must  have  two  terms  dis- 
tributed in  the  premisses,  namely,  the  middle  and  the  major 
(rules  3,  4).     But  a  particular  negative  proposition  and  a 


CHAP.  I.] 


SYLLOGISMS. 


145 


particular  affirmative  proposition  between    them  distribute 
only  one  term.     Therefore,  no  conclusion  can  be  obtained. 
[De  Morgan  {Forfnal  Logic,  p.  14)  proves  this  corollary 
as  follows  :— "  Since  both  premisses  are  particular  in  form, 
the  middle  term  can  only  enter  one  of  them  universally  by 
being  the  predicate  of  a  negative  proposition ;  consequently 
the  other  premiss  must  be  affirmative,  and,  being  particular, 
neither  of  its  terms  is  universal.      Consequently  both  the 
terms  as  to  which  the  conclusion  is  to   be   drawn   enter 
partially,  and  the  conclusion  can  only  be  a  particular  affir- 
mative proposition.    But  if  one  of  the  premisses  be  negative, 
the  conclusion  must  be  negative.     This  contradiction  shews 
that  the  supposition  of  particular  premisses   producing  a 
legitimate  result  is  inadmissible."] 

(ii)  If  one  premiss  is  particular,  so  must  be  the  conclusio7i\ 
We  must  have  either 

(a)    two  negative  premisses,  but  this  case  is  rejected 
by  rule  5 ; 

or       (/?)    two  affirmative  premisses  ; 

or       (y)     one  affirmative  and  one  negative. 

In  case  (^)  the  premisses,  being  both  affirmative  and 
one  of  them  particular,  can  distribute  but  one  term  between 
them.  This  must  be  the  middle  term  by  rule  3.  The  minor 
term  is  therefore  undistributed  in  the  premisses,  and  the 
conclusion  must  be  particular  by  rule  4. 

In  case  (y)  the  premisses  will  between  them  distribute 
two  and  only  two  terms.      These  must  be  the  middle  by 


^  This  and  the  sixth  rule  are  sometimes  combined  into  the  one  rule, 
Condusio  seqidfiir  partem  dcteyiorcm,—i.c.,  the  conclusion  follows  the 
worse  or  weaker  premiss  both  in  quality  and  in  quantity;  a  negative 
being  considered  weaker  than  an  affirmative,  and  a  particular  than  a 
universal. 


K.  L. 


10 


146 


SYLLOGISMS. 


[part  III. 


rule  3,  and  the  major  by  rule  4,  (since  we  have  a  negative 
premiss,  necessitating  a  negative  conclusion  by  rule  6, 
and  therefore  the  distribution  of  the  major  term  in  the 
conclusion).  Again,  therefore,  the  minor  cannot  be  dis- 
tributed in  the  premisses,  and  the  conclusion  must  be  par- 
ticular by  rule  4. 

[De  Morgan  {Formal  Logic,  ^;).  14)  gives  the  following 
very  ingenious  proof  of  this  corollary:—"  If  two  propositions 
P  and  (2,  together  prove  a  third,  R,  it  is  plain  that  P  and 
the  denial  of  R,  prove  the  denial  of  (2-  For  P  and  Q  can- 
not be  true  together  without  R.  Now  if  possible,  let  P 
(a  particular)  and"  Q  (a  universal)  prove  R  (a  universal). 
Then  P  (particular)  and  the  denial  of  R  (particular)  prove 
the  denial  of  Q.     But  two  particulars  can  prove  nothing."] 

(iii)  Fro7n  a  particular  major  and  a  negative  minor 
nothing  can  be  i?iferrcd. 

Since  the  minor  premiss  is  given  negative,  the  major 
premiss  must  by  rule  5  be  affirmative.  But  it  is  also  particular, 
and  it  therefore  follows  that  the  major  term  cannot  be  distri- 
buted in  it.  Hence,  by  rule  4,  it  must  be  undistributed  in 
the  conclusion,  />.,  the  con-elusion  must  be  affinnative.  But 
also  by  rule  6,  since  we  have  a  negative  premiss,  it  must 
be  7iemtive.  This  contradiction  establishes  the  corollary 
that  under  the  supposed  circumstances  no  conclusion  is 
possible. 

115.  Shew  by  aid  of  the  syllogistic  rules  that 
the  premisses  of  a  syllogism  must  contain  one  more 
distributed  term  than  the  conclusion ;  also,  that 
there  is  always  the  same  number  of  distributed  terms 
in  the  predicates  of  the  premisses  taken  together  as 
in  the  predicate  of  the  conclusion.     Hence  deduce 


CHAP.  I.] 


SYLLOGISMS. 


147 


the  three    corollaries.      [Cf.    Monck,    Ifitrodiiction   to 
Logic,  pp.  40,  41.] 

116.  "When  one  of  the  premisses  is  Particular, 
the  conclusion  must  be  Particular.  The  transgression 
of  this  rule  is  a  symptom  of  illicit  process  of  the 
minor."  Spalding,  Logic,  p.  209.  Is  it  the  case  that 
we  cannot  infer  a  universal  conclusion  from  a  parti- 
cular premiss  without  committing  the  fallacy  of  illicit 
minor  .^ 

117.  Illustrate  De  Morgan's  statement  that  any 
case  which  falls  under  the  rule  that  *'  from  premisses 
both  negative  no  conclusion  can  be  inferred"  may  be 
reduced  to  a  breach  of  one  of  the  preceding  rules. 

De  Morgan  {Formal  Logic,  p.  13)  takes  two  universal 
negative  premisses  E,  E,  In  whatever  figure  they  are,  they 
can  be  reduced  by  conversion  to, — 

No  P  is  M, 
No  6"  is  i/: 

Then  by  obversion  they  become,  (without  losing  any  of 
their  force), — 

All  P  is  not-^/, 
All  S  is  wol-M) 

and  we  have  undistributed  middle.     Hence  rule  5  is  ex- 
hibited as  a  corollary  from  rule  3. 

An  objection  may  perhaps  be  taken  to  the  above  on 
the  ground  that  the  premisses  might  also  be  reduced  to, — 

All  M  is  not-T', 

All  J/ is  not-^"; 

where  the  middle  term  is  distributed  in  both  premisses.  Here 

however  it  is  to  be  noted  that  we  have  no  longer  a  middle 

10 — 2 


148 


SYLLOGISMS. 


[part  III. 


term  coimecting  S  and  P  at  all.   We  shall  return  subsequently 
to  this  method  of  dealing  with  two  negative  premisses. 

The  case  in  which  one  of  the  premisses  is  particular  is 
dealt  with  by  De  Morgan  {Formal Logic,  p.  14)  as  follows: — 
"Again,  No  Fis  X,  Some  Fs  are  not  Zs,  may  be  converted 

into 

Every  X  is  (a  thing  which  is  not  K), 
Some  (things  which  are  not  Zs)  are  Fs, 

in  which  there  is  no  middle  term." 

This  is  not  quite  satisfactory,  since  we  may  often  exhibit 
a  valid  syllogism  in  such  a  form  that  there  appear  to  be 
four  terms ;  e.g.^  I  might  say,  "  All  M  is  P,  All  5  is  J/,  may 
be  converted  into 

All  M  is  P, 

No  S  is  not- J/, 

in  which  there  is  no  middle  term." 

The  case  in  question  may  however  be  disposed  of  by 
saying  that  if  we  can  infer  nothing  from  two  universal 
negative  premisses,  a  fortiori  we  cannot  from  two  negative 
premisses,  one  of  which  is  particular. 

118.  The  rule  that  "  if  one  premiss  Is  negative, 
the  conclusion  must  be  negative,"  may  be  established 
as  a  corollary  from  the  rule  that  "  from  two  negative 
premisses  nothing  can  be  inferred." 

The  following  has  been  suggested  to  me  by  Dc  Morgan's 
deduction  of  corollary  ii.,  (cf.  section  114): — If  two  pro- 
positions P  and  Q  together  prove  a  third  R,  it  is  plain  that 
P  and  the  denial  of  R  prove  the  denial  of  Q.  For  P  and  Q 
cannot  be  true  together  without  R.  Now  if  possible  let  P 
(a  negative)  and  Q  (an  affirmative)  prove  R  (an  affirmative). 
Then  P  (a  negative)  and  the  denial  of  R  (a  negative)  prove 
the  denial  of  Q,     But  two  negatives  prove  nothing. 


CHAP.  I.] 


SYLLOGISMS. 


149 


119.  Simplification  of  the  Rules  of  the  Syllogism. 
It  would  now  seem  as  if  the  six  rules  of  the  syllogism 

might  be  simplified.  Rules  i  and  2  may  be  treated  as 
a  description  of  the  syllogism  rather  than  as  rules  for  its 
validity.  The  part  of  rule  3  relating  to  ambiguity  may  be 
regarded  as  contained  in  the  proviso  that  there  shall  be  only 
three  terms,  (i.e.,  if  one  of  the  terms  is  ambiguous,  we 
have  not  really  a  syllogism  according  to  our  definition  of 
syllogism).  Rule  5  has  been  exhibited  in  section  117  as 
a  corollary  from  rule  3 ;  and  the  first  part  of  rule  6  has 
been  shewn  in  section  118  to  be  a  corollary  from  rule  5. 
We  are  left  then  with  only  three  independent  rules,— 

(a)  The  middle  term  must  be  distributed  once  at  least 
in  the  premisses ; 

(/3)  No  term  must  be  distributed  in  the  conclusion  un- 
less it  has  been  distributed  in  the  premisses ; 

(y)  A  negative  conclusion  cannot  be  inferred  from  two 
affirmative  premisses. 

120.  In  reference  to  the  syllogism,  it  has  been 
urged  that  the  old  rule  that  negative  premisses  yield 
no  conclusion  does  not  hold  true  universally,  as  in 
the  example,  Whatever  is  not  metallic  is  not  capable 
of  powerful  magnetic  influence,  carbon  is  not  metallic, 
therefore,  carbon  is  not  capable  of  powerful  magnetic 
influence.     Examine  this  criticism.  [c] 

Professor  Jevons  gives  this  case  in  his  Principles  of 
Science  (ist  edition,  vol.  i.,  p.  76;  2nd  edition,  p.  (i2>\  and  he 
states  that  "the  syllogistic  rule  is  actually  falsified  in  its 
bare  and  general  statement." 

Professor  Croom  Robertson  has  however  conclusively 
shewn  (in  Mind,  1876,  p.  219,  7iote)  that  this  apparent  ex- 


ISO  SYLLOGISMS.  [part  in. 

ception  is  no  real  exception'.  **  There  OirQ/onr  terms  in  the 
example,  and  thus  no  syllogism,  if  the  premisses  are  taken 
as  negative  propositions;  while  the  minor  premiss  is  an  affir- 
ifiative  proposition,  if  the  terms  are  made  of  the  requisite 
number  three." 

Mr  Bradley  {Principles  of  Logic,  p.  254)  returns  to  the 
position  taken  by  Professor  Jevons.  In  reference  to  the 
example  given  in  the  above  question,  he  says,  "  This  argu- 
ment no  doubt  has  quaternio  ierminorum  and  is  vicious 
technically,  but  the  fact  remains  that  from  two  denials  you 
somehow  have  proved  a  further  denial.  *  A  is  not  B,  what 
is  not  B  is  not  C,  therefore  A  is  not  C;  the  premisses  are 
surely  negative  to  start  with,  and  it  appears  pedantic  either 
to  urge  on  one  side  that  'A  is  not-^'  is  simply  positive,  or 
on  the  other  that  B  and  not--^  afford  no  junction.  If  from 
negative  premisses  I  can  get  my  conclusion,  it  seems  idle  to 
object  that  I  have  first  transformed  one  premiss  ;  for  that 
objection  does  not  shew  that  the  premisses  are  not  negative, 
and  it  does  not  shew  that  I  have  failed  to  get  my  con- 
clusion." 

This  is  somewhat  beside  the  mark ;  and  if  the  points 
on  both  sides  are  clearly  stated  there  appears  no  room  for 
further  controversy.  On  the  one  hand,  it  is  implicitly 
admitted  both  by  Professor  Jevons  {Studies  in  Deductive 
Logic,  p.  89),  and  by  Mr  Bradley,  that  two  negative 
premisses  invalidate  a  syllogism,  i.e.,  understanding  by  a 
syllogism  a  mediate  reasoning  containing  three  and  only 
three  terms.  On  the  other  hand,  everyone  would  allow  that 
from   two  propositions  which  may  both   be  regarded  as 


^  Mr  Venn,  also,  (in  the  Academy,  Oct.  3,  1874), — "The  reply 
clearly  is,  that  if  '  not  metallic'  is  to  be  regarded  as  the  predicate  of  the 
minor,  then  the  minor  is  affirmative;  if  'metallic'  is  predicate,  then 
there  are  four  terms." 


CHAP.  1.] 


SYLLOGISMS. 


151 


negative,  a  conclusion  may  sometimes  be  obtained;  for 
example,  the  propositions  which  constitute  the  premisses  of 
a  syllogism  in  Barbara^  may  be  written  in  a  negative  form, 
thus.  No  M  is  not-P,  No  S  is  not- J/,  and  no  doubt  the  con- 
clusion— All  S  is  /'—still  follows.  We  must  not,  however, 
attach  undue  importance  to  the  distinction  between  positive 
and  negative  propositions.  By  means  of  the  process  of  Ob- 
version,  the  logician  may  at  will  regard  any  given  propo- 
sition as  cither  positive  or  negative. 

[A  similar  case  to  that  given  in  the  question  is  dealt 
with  in  the  Port  Royal  Logic  (Professor  Baynes's  translation, 
p.  211)  as  follows  : — 

**  There  are  many  reasonings,  of  which  all  the  pro- 
positions appear  negative,  and  which  are,  nevertheless,  very 
good,  because  there  is  in  them  one  which  is  negative  only 
in  appearance,  and  in  reality  affirmative,  as  we  have  already 
shewn,  and  as  we  may  still  further  see  by  this  example : 

That  which  has  no  parts  cannot  perish  by  the  dissolution 
of  its  parts  ; 

The  soul  has  ?io  parts; 

Therefore,  the  soul  cannot  perish  by  the  dissolution  of  its 
parts. 

There  are  several  who  advance  such  syllogisms  to  shew 
that  we  have  no  right  to  maintain  unconditionally  this 
axiom  of  logic,  Nothifig  can  be  inferred  from  pure  negatives; 
but  they  have  not  observed  that,  in  sense,  the  minor  of  this 
and  such  other  syllogisms  is  affirmative,  since  the  middle, 
which  is  the  subject  of  the  major,  is  in  it  the  attribute.  Now 
the  subject  of  the  major  is  not  that  which  has  parts,  but 


1  KWMx^P, 

All  S  is  M, 
therefore,  All  6"  is  P.     Cf.  section  158. 


152 


SYLLOGISMS. 


[part  III. 


that  which  has  not  parts,  and  thus  the  sense  of  the  minor  is, 
The  soul  is  a  thing  without  parts,  which  is  a  proposition 
affirmative  of  a  negative  attribute."] 

121.  By  what  means  can  we  obtain  a  conclusion 
from  the  two  negative  premisses, — 

No  M  is  P, 
No  J/ is  5? 

By  obverting  the  premisses,  we  have — , 

All  ^1/ is  not-/', 
All  J/ is  not-^S", 
therefore,  Some  not-^Sis  not-/**. 

122.  Take  an  apparent  syllogism  subject  to  the 
fallacy  of  negative  premisses,  and  enquire  whether 
you  can  correct  the  reasoning  by  converting  one  or 
both  of  the  premisses  into  the  affirmative  form.  [Je- 
vons,  Studies  in  Deductive  Logic ^  p.  84.] 

Both  in  the  Studies  and  in  the  Principles  of  Science  (Vol. 
I.,  p.  75),  Professor  Jevons  appears  to  answer  this  question  in 
the  negative.  It  is  certainly  not  put  in  an  unexceptionable 
form,  but  apparently  reference  is  made  to  the  case  given  in 
the  preceding  section. 

No  A  is  B, 

No  A  is  C, 

may  be  transformed  into, — 

All  A  is  not-i?, 
All  A  is  not-C; 

^  But  this  does  not  invalidate  the  syllogistic  rule  that  from  two  nega- 
tive premisses  nothing  can  be  inferred,  since  so  long  as  both  the  pre- 
misses remain  negative  we  have  more  than  three  terms  and  therefore 
not  a  syllogism  at  all. 


CHAP.  I.]  SYLLOGISMS.  i53 

yielding  a  conclusion, — 

Some  not- C  is  not-^. 
[In  Jevons's  system,  this  would  become, — 

A  =  Ab, 

A^AC'y 

yielding  a  conclusion, — 

Ab  -  Ac. 

(Cf  Principles  of  Science,  vol.  i.,  p.  71;  2nd  ed.,  p.  59).] 

123.     Given 

(i)     All  P  is  M, 
(ii)    All  5  is  M, 

(iii)     M  does  not  constitute  the  entire  universe 
of  discourse.     What  conclusion  can  we  infer  ? 

Exhibit  the  reasoning  in  the  form  of  an  Aristote- 
lian syllogism. 

Is  the  third  premiss  necessary  in  order  that  the 
conclusion  may  be  obtained  t  Make  any  comments 
that  occur  to  you  in  connection  with  this  point. 

From  (i)  we  can  obtain  by  immediate  inference.  All 
not- J/  is  not-/;  and  from  (ii)  All  not- J/  is  not-^" ;  and  these 
premisses  yield  the  conclusion, — 

Some  wo\.-S  is  not-/*. 
The  reasoning  is  here  exhibited  in  the  form  of  an  Aristote- 
lian syllogism. 

Or,  we  might  reason  as  follows :— Since  S  and  P  are 
both  entirely  included  in  M,  there  must  be  outside  M  some 
not-5  and  some  not-P  that  are  coincident ;  and  this  is  the 
same  conclusion  as  before. 

Now  in  the  latter  form  of  the  reasoning  it  would  seem 
that  we  have  assumed  that  there  is  some  not- J/,  i.e.,  that  M 


154 


SYLLOGISMS. 


[part  III. 


does  not  constitute  the  entire  universe  of  discourse.  But 
the  necessity  of  this  assumption  was  not  apparent  in  our 
first  method  of  treatment,  according  to  which  by  a  simple 
process  of  immediate  inference  we  obtained  a  perfectly 
valid  syllogism  ^ 

The  truth  appears  to  be  that  here  at  any  rate  we  have 

an  illustration  of  De  Morgan's  view  {Formal  Logic,  p.  112) 

that  in  all  syllogisms  the  existence  of  the  middle  term  is  a 

datum.     From  the  premisses  All  M  is  P,  All  M  is  S,  we 

cannot  obtain  the  conclusion  Some  S\%P  without  implicitly 

assuming  the  existence  of  J/.     Take  as  an  example,— All 

witches  ride  through  the  air  on  broomsticks;  All  witches  are 

old  women;  therefore,  Some  old  women  ride  through  the  air 

on  broomsticks.     This  point  is  further  discussed  in  sections 

273—277- 

We  may  note  that  the  reasoning, — 

All  P  is  M, 
All  6*  is  M, 
therefore.  Some  not-^  is  not-/", 
does  not  invalidate  the  syllogistic  rule  that  the  middle  term 
must  be  distributed  once  at  least  in  the  premisses,  since  as 
it  stands  it  contains  more  than  three  terms  and  is  therefore 
not  a  syllogism. 

124.  Examine  the  following  assertion:  "In  no 
way  can  a  syllogism  with  two  singular  premisses  be 
viewed  as  a  genuine  syllogistic  or  deductive  inference." 

[W.] 

This  assertion  is  made  by  Professor  Bain,  and  he  illus- 
trates it  {Logic,  Deduction,  p.  159)  by  reference  to  the  fol- 
lowing syllogism  ; 


^  Compare,  however,  section  104. 


CHAP.  I.] 


SYLLOGISMS. 


155 


Socrates  fought  at  Delium, 
Socrates  was  the  master  of  Plato, 
therefore,  The  master  of  Plato  fought  at  Delium. 

But  *'the  proposition  'Socrates  was  the  master  of  Plato 
and  fought  at  Delium ',  compounded  out  of  the  two  pre- 
misses is  nothing  more  than  a  grammatical  abbreviation  " ; 
and  the  step  hence  to  the  conclusion  is  a  mere  omission  of 
something  that  had  previously  been  said.  "  Now,  we  never 
consider  that  we  have  made  a  real  'nference,  a  step  in  ad- 
vance, when  we  repeat  less  than  we  are  entitled  to  say,  or 
drop  from  a  complex  statement  some  portion  not  desired 
at  the  moment.  Such  an  operation  keeps  strictly  within  the 
domain  of  ^Equivalence  or  Immediate  Inference.  In  no 
way,  therefore,  can  a  syllogism  with  two  singular  premisses  be 
viewed  as  a  genuine  syllogistic  or  deductive  inference." 

The  above  leads  up  to  some  very  interesting  considera- 
tions, but  it  proves  too  much.  In  the  following  syllogisms 
the  premisses  may  be  similarly  compounded  together, — 

all  men  are  mortal,  )    „  .  1      j     .•       i 

.       ,  >  all  men  are  mortal  and  rational ; 
all  men  are  rational,) 

therefore,  some  rational  beings  are  mortal. 

all  men  are  mortal, V 
all  kings  are  men,  j 
therefore,  all  kings  are  mortal  \ 


all  men  including  kings  are  mortal ; 


^  With  the  above,  compare  the  following  syllogism,  having  two 
singular  premisses : — 

The  Lord  Chancellor  receives  a  higher  salary  than  the  Prime 
Minister, 

Lord  Selborne  is  the  Lord  Chancellor, 

therefore.  Lord  Selborne  receives  a  higher  salary  than  the  Prime 
Minister. 

The  premisses  here  would  similarly,  I  suppose,  be  compounded  by 
Professor  Bain  into  "The  Lord  Chancellor,  Lord  Selborne,  receives  a 
higher  salary  than  the  Prime  Minister." 


156 


SYLLOGISMS. 


[part  III. 


Do  not  Bain's  criticisms  apply  to  these  syllogisms  as 
much  as  to  the  syllogism  with  two  singular  premisses  ?  The 
method  of  treatment  adopted  is  indeed  particularly  ap- 
plicable to  syllogisms  in  which  the  middle  term  is  subject 
in  both  premisses  ^;  but  in  any  case  it  is  true  that  the  con- 
clusion of  a  syllogism  contains  a  part  of,  and  only  a  part  of, 
the  information  contained  in  the  two  premisses  taken  to- 
gether. Also,  we  may  always  combine  the  two  premisses  in 
a  single  statement;  and  thus  we  may  always  get  Bain's 
result.  In  other  words,  in  the  conclusion  of  every  syllogism 
*Sve  repeat  less  than  we  are  entitled  to  say,"  or,  if  we  care  to 
put  it  so,  *'drop  from  a  complex  statement  some  portion 
not  desired  at  the  moment." 

It  may  be  worth  while  here  to  refer  to  the  charge 
of  incompleteness  which  Professor  Jevons  (^Principles  of 
Science,  i.  p.  71)  has  brought  against  the  ordinary  syllogistic 
conclusion.  *' Potassium  floats  on  water,  Potassium  is  a 
metal,*'  yield,  according  to  him,  the  conclusion,  "Potassium 
metal  is  potassium  floating  on  water."  But  "Aristotle 
would  have  inferred  that  some  metals  float  on  water.  Hence 
Aristotle's  conclusion  simply  leaves  out  some  of  the  informa- 
tion afforded  in  the  premisses;  it  even  leaves  us  open  to 
interpret  the  some  metals  in  a  wider  sense  than  we  are 
warranted  in  doing." 

In  reply  to  this  it  may  be  remarked :  first,  that  the  Aris- 
totelian conclusion  does  not  profess  to  sum  up  the  whole 
of  the  information  contained  in  the  premisses  of  the  syl- 
logism; secondly,  that  some  in  Logic  means  merely  *'not 
none",  "one  at  least".  The  conclusion  of  the  above  syllo- 
gism might  perhaps  better  be  written  "some  metal  floats  on 
water,"  or  "some  metal  or  metals,  &c."    Compare  Mr  Venn*, 

^  i.e.^  to  syllogisms  in  Figure  3.     Cf.  section  143. 

^  *' Surely,  as  the  old  expression  'discursive  thought'  implies,  we 


CHAP.  I.] 


SYLLOGISMS. 


157 


in  the  Academy^  Oct.  3,  1874;  also,  Professor  Croom  Robert- 
son in  Mind,  1876,  p.  219. 

125.  How  far  does  the  conclusion  of  an  Aristo- 
telian syllogism  fall  short  of  giving  all  the  informa- 
tion contained  in  the  premisses }  [Jevons,  Studies, 
p.  215.] 

126.  The  connection  between  the  Dictum  de  omui 
ct  nullo  and  the  ordinary  rules  of  syllogism. 

The  Dictum  de  omni  et  nullo  was  given  by  Aristotle 
as  the  axiom  on  which  all  syllogistic  inference  is  based.  It 
applies  directly,  however,  to  those  syllogisms  only  in  which 
the  major  term  is  predicate  in  the  major  premiss,  and  the 
minor  term  subject  in  the  minor  premiss,  (i.e.,  to  what  are 
called  syllogisms  in  Figure  i).  The  rules  of  syllogism,  on 
the  other  hand,  apply  independently  of  the  position  of  the 
terms  in  the  premisses.  Nevertheless,  it  is  interesting  to 
trace  the  connection  between  them.  We  shall  find  all  the 
rules  implicitly  contained  in  the  Dictum,  but  some  of  them 
in  a  less  general  form,  in  consequence  of  the  distinction 
pointed  out  above. 

The  Dictum  may  be  stated  as  follows : — "Whatever  is 
predicated,  whether  affirmatively  or  negatively,  of  a  term 
distributed  may  be  predicated  in  like  manner  of  everything 
contained  under  it." 


designedly  pass  on  from  premisses  to  conclusion,  and  then  drop  the 
premisses  from  sight.  If  we  want  to  keep  them  in  sight  we  can  perfectly 
well  retain  them  as  premisses ;  if  not,  if  all  that  we  want  is  the  final  fact, 
it  is  no  use  to  burden  our  minds  or  paper  with  premisses  as  well  as  con- 
clusion. All  reasoning  is  derived  from  data  which  under  conceivable 
circumstances  might  be  useful  again,  but  which  we  are  satisfied  to  recover 
when  we  want  them." 


158  SYLLOGISMS.  [part  in. 

(i)  The  Dictum  provides  for  three  and  only  three  terms ; 
namely,  (i)  a  certain  term  which  must  be  distributed,  (ii) 
something  predicated  of  this  term,  (iii)  something  contained 
under  it.  These  terms  are  respectively  the  middle,  major, 
and  minor.  We  may  consider  the  rule  relating  to  the 
ambiguity  of  terms  also  contained  here,  since  if  any  term  is 
ambiguous  we  have  practically  more  than  three  terms. 

(2)  The  Dictum  provides  for  three  and  only  three  pro- 
positions; namely,  (i)  a  proposition  predicating  something 
of  a  term  distributed,  (ii)  a  proposition  declaring  something 
to  be  contained  under  this  term,  (iii)  a  proposition  making 
the  original  predication  of  the  contained  term.  These  pro- 
positions constitute  respectively  the  major  premiss,  the  minor 
premiss,  and  the  conclusion  of  the  syllogism. 

(3)  The  Dictum  prescribes  not  merely  that  the  middle 
term  shall  be  distributed  once  at  least  in  the  premisses,  but 
more  explicitly  that  it  shall  be  distributed  in  the  major 
premiss, — "Whatever  is  predicated  of  a  term  distributed^ 
[This  is  really  another  form  of  what  we  shall  find  to  be  a 
special  rule  of  Figure  i,  namely  that  the  major  premiss  must 
be  universal.     Cf  section  144.] 

(4)  The  proposition  declaring  that  something  is  con- 
tained under  the  term  distributed  must  necessarily  be  an 
affirmative  proposition.  The  Dictum  provides  therefore  that 
the  premisses  shall  not  be  both  negative.  [It  really  provides 
that  the  minor  premiss  shall  be  affirmative,  which  again  is 
one  of  the  special  rules  of  Figure  i.] 

(5)  The  words  *'in  like  manner"  clearly  provide  against 
a  breach  of  rule  6,  namely  that  if  one  premiss  is  negative, 
the  conclusion  must  be  negative,  and  vice  versa. 

(6)  Illicit  process  of  the  major  is  provided  against  indi- 
rectly.    We  can  commit  this  fallacy  only  if  we  have  a  nega- 


CHAP.  I.] 


SYLLOGISMS. 


159 


.  1 


tive  conclusion,  but  the  words  "in  like  manner"  declare 
that  if  we  have  a  negative  conclusion,  we  must  have  a  nega- 
tive major  premiss,  and  since  in  any  syllogism  to  which  the 
Dictum  directly  applies,  the  major  term  is  predicate  of  this 
premiss,  it  likewise  will  be  distributed. 

Illicit  process  of  the  minor  is  simply  provided  against 
inasmuch  as  we  are  warranted  to  make  our  predication  in 
the  conclusion  only  of  what  has  been  shewn  in  the  minor 
premiss  to  be  contained  under  the  middle  term. 

127.  Can  the  Syllogism  be  based  exclusively  on 
the  laws  of  Identity,  Contradiction  and  Excluded 
Middle  > 

Mansel  answers  this  question  in  the  affirmative  and  main- 
tains {Prolegomena  Logica,  p.  222)  that  ''the  Principle  of 
Identity  is  immediately  applicable  to  affirmative  moods  in 
any  figure,  and  the  Principle  of  Contradiction  to  negatives." 
In  order  to  shew  this,  he  commences  by  quantifying  the 
predicate  (cf  section  217),  and  taking  as  an  example  the 
syllogism, — 

All  M  is  some  P, 

All  S  h  some  J/, 
therefore.  All  S  is  some  P^ 

he  reads  it  thus,—  "  the  minor  term  all  ^  is  identical  with  a 
part  of  J^  and  consequently  with  a  part  of  that  which  is 
given  as  identical  with  all  M,  namely  some  P:'  He  then 
takes  the  syllogism, — 

All  Af  is  some  P, 
Some  S  is  some  J/, 
therefore.  Some  S  is  some  P, 

and,  treating  it  similarly,  finds  that  "the  principle  Immedi- 
ately applicable  to  both  is  the  axiom,  that  what  is  given  as 
identical  with  the  whole  or  a  part  of  any  concept,  must  be 


i6o 


SYLLOGISMS. 


[part  III. 


identical  with  the  whole  or  a  part  of  that  which  is  identical 
with  the  same  concept."  Passing  by  the  inaccuracy  of 
speaking  of  the  concepts  as  being  identical*,  I  cannot  see  that 
the  above  axiom  is  the  same  as  the  Principle  of  Identity, 
"Every  A  is  -<4."  The  syllogism  is  something  more  than 
mere  subaltern  inference  \  it  involves  a  passage  of  thought 
through  a  middle  term ;  and  it  is  just  this  that  the  Law  of 
Identity  as  expressed  in  the  formula  "  Every  A  is  A'^  ap- 
pears to  me  unable  to  provide  for. 

This  law  may  tell  us  that  if  all  M  is  /*,  then  some  M  is 
P;  but  does  it  tell  us  that  if  all  J/ is  F,  therefore  S  is  Pj 
because  it  is  J/?  The  Dictum  de  omni  et  nulla  clearly 
enunciates  the  principle  involved  in  syllogistic  reasoning; 
the  Law  of  Identity,  if  it  does  so  at  all,  does  so  less  satis- 
factorily. Or  rather  I  would  say  that  if  the  Law  of  Identity 
is  to  cover  this  principle,  then  it  is  inadequately  expressed 
in  the  formula  Every  A  \s  A^.  Similar  remarks  apply  to  the 
attempt  to  bring  syllogisms  with  negative  conclusions  under 
the  Principle  of  Contradiction,  "No  A  is  not-^." 


^  It  is  really  the  extension  of  the  one  concept  that  is  identical  with 
the  whole  or  a  part  of  the  extension  of  the  other;  and  although  the 
comprehension  of  a  concept  is  practically  the  concept  itself,  it  is  clear 
that  the  same  is  not  true  of  its  extension.  It  has  always  seemed  to 
me  rather  curious  that  the  doctrine  of  the  Quantification  of  the  Predicate 
should  have  been  introduced  by  writers  like  Hamilton  and  Mansel, 
who  lay  so  much  stress  on  concepts. 

2  I  should  say  the  same  in  reference  to  Mansel's  remark  {Prole- 
gomena Logica,  p.  103),  that  the  Axiom  "things  that  are  equal  to  the 
same  are  equal  to  one  another"  is  only  another  statement  of  the 
Principle  of  Identity. 


CHAPTER  IL 


SIMPLE   EXERCISES   ON   THE   SYLLOGISM. 


128.  Explain  what  is  meant  by  a  Syllogism;  and 
put  the  following  argument  into  syllogistic  form  :— 
^*  We  have  no  right  to  treat  heat  as  a  substance,  for 
it  may  be  transformed  into  something  which  is  not 
heat,  and  is  certainly  not  a  substance  at  all,  namely, 
mechanical  work."  r^i 

129.  Put  the  following  argument  into  syllogistic 
form:  —  How  can  any  one  maintain  that  pain  is 
always  an  evil,  who  admits  that  remorse  involves 
pain,  and  yet  may  sometimes  be  a  real  good  }    fv.] 

130.  It  has  been  pointed  out  by  Ohm  that 
reasoning  to  the  following  effect  occurs  in  some 
works  on  mathematics: — '*A  magnitude  required  for 
the  solution  of  a  problem  must  satisfy  a  particular 
equation,  and  as  the  magnitude  x  satisfies  this  equa- 
tion, it  is  therefore  the  magnitude  required." 

I'Lxamine  the  logical  validity  of  this  argument,  [c] 

131.     If  P  is  a  mark  of  the  presence  of  Q,  and  R 
of  that  of  6',  and  if  P  and  R  are  never  found  together. 


K.  L. 


II 


SYLLOGISMS. 


[part  III. 


162 

am  I  right  in  inferring  that  Q  and  5  sometimes  exist 
separately  ?  l^-J 

The  premisses  may  be  stated, — 

All  P  is  (2, 
All  R  is  5, 
No  /^  is  i? ; 

and  in  order  to  establish  the  desired  conclusion  we  must 
be  able  to  infer  at  least  one  of  the  following, — 

Some  Q  is  not  5, 

Some  S  is  not  Q. 

But  neither  of  these  propositions  can  be  inferred,  since 
they  distribute  respectively  S  and  Q,  whilst  neither  of  these 
terms  is  distributed  in  the  given  premisses.  The  question 
is  therefore  to  be  answered  in  the  negative. 

132.  If  it  is  false  that  the  attribute  B  is  ever 
found  coexisting  with  A,  and  not  less  false  that  the 
attribute  C  is  sometimes  found  absent  from  A,  can 
you  assert  anything  about  B  in  terms  of  d        [c] 

133.  Enumerate  the  cases  in  which  no  valid  con- 
clusion can  be  drawn  from  two  premisses. 

134.  Shew  that 

(i)  If  both  premisses  of  a  syllogism  arc  affirma- 
tive, and  one  but  only  one  of  them  universal,  they 
will  between  them  distribute  only  one  term ; 

(ii)  If  both  premisses  are  affirmative  and  both 
universal,  they  will  between  them  distribute  two 
terms; 

(iii)     If  one   but  only  one  premiss  is  negative, 


CHAP.  II.] 


SYLLOGISMS. 


163 


and  one  but  only  one  premiss  universal,  they  will 
between  them  distribute  two  terms; 

(iv)  If  one  but  only  one  premiss  is  negative, 
and  both  premisses  arc  universal,  they  will  between 
them  distribute  three  terms. 

135.  Ascertain  how  many  distributed  terms  there 
may  be  in  the  premisses  of  a  syllogism  more  than 
in  the  conclusion.  ^^1 

^  136.     Prove  that,  when  the  minor  term  is  predicate 
in  its  premiss,  the  conclusion  cannot  be  A.  [l.] 

137.  If  the  major  term  of  a  syllogism  be  the 
predicate  of  the  major  premiss,  what  do  we  know 
about  the  minor  premiss  }  ["l  i 

138.  How  much  can  you  tell  about  a  valid 
syllogism  if  you  know, — 

(i)     that  only  the  middle  term  is  distributed; 

(2)  that  only  the  middle  and  minor  terms  are 
distributed; 

(3)  that  all  three  terms  are  distributed  ?      [w.] 

139.  If  it  be  known  concerning  a  syllogism  in 
the  Aristotelian  system  that  the  middle  term  is  dis- 
tributed in  both  premisses,  what  can  we  infer  as  to 
the  conclusion  }  r^i 

If  both  premisses  are  affirmative,  they  can  between 
them  distribute  only  two  terms;  but  by  hypothesis  the 
middle  term  is  distributed  twice  in  the  premisses,  the  minor 
term  cannot  therefore  be  distributed,  and  it  follows  that  the 
conclusion  must  be  particular. 

II — 2 


164. 


SYLLOGISMS. 


[part  III. 


If  one  of  the  premisses  is  negative,  we  may  have  three 
terms  distributed  in  the  premisses ;  these  must,  however,  be 
the  middle  term  twice  (by  hypothesis),  and  the  major  term 
(since  the  conclusion  must  now  be  negative  and  the  major 
term  will  therefore  be  distributed  in  it);  hence  the  minor 
term  cannot  be  distributed  in  the  premisses,  and  it  again 
follows  that  the  conclusion  must  be  particular. 

But  either  both  premisses  will  be  affirmative,  or  one 
affirmative  and  the  other  negative ;  in  any  case,  therefore, 
we  can  infer  that  the  conclusion  will  be  particular. 

[This  proof  seems  preferable  to  that  given  by  Jcvons, 
Studies  in  Deductive  Logic,  p.  83.] 

140.  Shew  that  if  the  conclusion  of  a  syllogism 
be  a  universal  proposition,  the  middle  term  can  be 
but  once  distributed  in  the  premisses.  [l.] 

As  pointed  out  by  Professor  Jevons  {Studies  in  Deductive 
Logic,  p.  85),  this  proposition  is  the  contrapositive  of  the 
result  obtained  in  the  preceding  section. 

141.  Shew  directly  in  how  many  ways  it  is  pos- 
sible to  prove  the  conclusions  SaP,  SeP;  point  out 
those  that  conform  immediately  to  the  Dictum  de  omni 
et  nidlo ;  and  exhibit  the  equivalence  between  these 
and  the  remainder.  [w.] 

(i)     To  prove  ^// 5 /V  P. 

Both  premisses  must  be  affirmative,  and  both  must  be 

universal. 

S  being  distributed  in  the  conclusion,  must  be  distri- 
buted in  the  minor  premiss,  which  must  therefore  be  All  S 

is  M, 

M  not  being  distributed  in  the  minor  must  be  distri- 
buted in  the  major  which  must  therefore  be  All  M is  F, 


CHAP.  II.]  SYLLOGISMS.  165 

SaP  can  therefore  be  proved  in  only  one  way,  namely. 

All  M  is  P, 
All  S  is  M, 
therefore,  All  S  is  p]^ 

and  this  syllogism  conforms  immediately  to  the  Diciufn. 

(2)     To  prove  No  S  is  P. 

Both  premisses  must  be  universal,  and  one  must  be 
negative  while  the  other  is  affirmative,  i.e.,  one  premiss  must 
be  E  and  the  other  A, 

First,  let  the  major  be  E,  i.e., 

either  A^  M  is  P  or  No  P  is  M, 

In  each  case  the  minor  must  be  affirmative  and  must  dis- 
tribute S',  therefore,  it  will  be  Ad  S  is  M. 

Secondly,  let  the  minor  be  E,  i.e., 

either  N'o  M  is  S  or  No  S  is  M. 

In  each  case  the  major  must  be  affirmative  and  must  dis- 
tribute P\   therefore,  it  will  be  Ad  P  is  M, 

We  can  then  prove  ScP  in  four  ways,  tluis, 

(i)  MeP,         (ii)  FeM,         (iii)  FaM,         (iv)  FaM, 
6Vz^  SaM,  MeS,  Scm] 


SeP, 


Sel\ 


SeP, 


SeP. 


Of  these,  (i)  only  conforms  immediately  to  the  Dictum, 
and  we  have  to  shew  the  equivalence  between  it  and  the 
others. 

The  only  difference  between  (i)  and  (ii)  is  that  the  major 
premiss  of  the  one  is  the  simple  converse  of  the  major 
premiss  of  the  other;  they  are  therefore  equivalent.  Simi- 
larly the  only  difference  between  (iii)  and  (iv)  is  that  the 
minor  premiss  of  the  one  is  the  simple  converse  of  the 


j_ 


i66 


SYLLOGISMS. 


[part  III. 


minor  premiss   '^f    the  other;   they  are  therefore  equiva- 
lent. 

Finally,  we  may  snew  that  (iii)  is  equivalent  to  (i)  by 
transposing  the  premisses  and  converting  the  conclusion. 

142.    Shew  directly  in  how  many  ways  it  is  possible 
to  prove  the  conclusions  SiP,  SoP,  [w.l 


CHAPTER  in. 


THE    FIGURES   AND    MOODS    OF   THE    SYLLOGISM. 


143.     Figure  and  Mood. 

By  the  Figure  of  a  Syllogism  is  meant  the  position  of 
the  terms  in  the  premisses. 

Denoting  the  major,  middle  and  minor  terms  by  the 
letters  P,  M,  S  respectively,  and  stating  the  major  premiss 
first,  \ye  have  four  figures  of  the  syllogism  as  shewn  in  the 
following  table: — 


^^'s:-  3' 


Fig.   I.  Fig.  2. 

M-P  P-M  M-P 

S-M  S-M  M-  S 

S-P  S-P  ~S-I> 


Fig.  4. 

P-M 
M~S 
S-P. 


By  the  Mood  of  a  Syllogism  is  meant  the  quantity  and 
quality  of  the  premisses  and  conclusion.  Thus  AAA,  EIO 
are  different  moods.  It  is  clear  that  if  figure  and  mood  are 
both  given,  the  syllogism  is  given. 

144.     The  Special  Rules  of  the  Figures  ;   and  the 

Determination    of    the    Legitimate   Moods   in   each 
Figure\ 

^  The  method  of  Determination  here  adopted  is  only  one  amongst 
several  possible  methods.  Another  is  suggested,  for  example,  in  sections 
141,  142. 


1 68 


SYLLOGISMS. 


[part  III. 


It  may  first  of  all  be  shewn  that  certain  combinations  of 
premisses  are  incapable  of  yielding  a  valid  conclusion  in 
any  figure.  A  priori^  there  are  possible  the  following  six- 
teen different  combinations  of  premisses,  the  major  premiss 
being  always  stated  first: — A  A,  A  I,  AE,  AO,  I  A,  21^  IE, 
10,  EA,  EI,  EE,  EO,  OA,  01,  OE,  00.  Referring  back 
however  to  the  syllogistic  rules  (section  114),  we  find  that  of 
these,  EE,  EO,  OE,  00,  (being  combinations  of  negative 
premisses),  give  no  conclusion  by  rule  5 ;  again,  //,  10,  01, 
(being  combinations  of  particular  premisses),  are  excluded 
by  corollary  i.;  and  IE  is  excluded  by  corollary  iii.,  which 
tells  us  that  nothing  follows  from  a  particular  major  and  a 
negative  minor. 

We  are  left  then  with  the  following  eight  possible  com- 
binations :—^^,  A  I,  AE,  AO,  I  A,  EA,  EI,  OA ;  and 
we  may  now  go  on  to  determine  in  which  figures  these  will 
yield  conclusions. 

T/ie  special  rules  and  the  legitimate  moods  of  Figure  i . 
The  position  of  the  terms  in  Figure  i  is  shewn  thus, — 

M-P 

S-P 

and  we  can  prove  that  in  this  figure : — 

(i)  The  minor  premiss  must  be  affij-mative.  For  if  it  were 
negative,  the  major  premiss  would  have  to  be  affirmative  by 
rule  5,  and  the  conclusion  negative  by  rule  6.  The  major 
term  would  therefore  be  distributed  in  the  conclusion,  and 
undistributed  in  its  premiss;  and  the  syllogism  would  be 
invalid  by  rule  4. 

(2)  The  major  premiss  must  be  universal.  For  the  middle 
term  cannot  be  distributed  in  the  minor  premiss  since  this 


CHAP.  III.] 


SYLLOGISMS. 


169 


is  affirmative,  and  must  therefore  be  distributed  in  the  major 
premiss. 

Rule  (i)  shews  that  AE  and  AO,  and  rule  (2)  that  lA 
and  OA  yield  no  conclusions  in  this  figure.  We  are  there- 
fore left  with  only  four  combinations,  namely,  A  A,  AI,  EA, 
EI  Applying  the  rules  that  a  negative  premiss  gives  a 
negative  conclusion,  while  conversely  a  negative  conclusion 
requires  a  negative  premiss,  and  that  a  particular  premiss 
gives  a  particular  conclusion  only,  we  find  that  AA  will 
justify  either  of  the  conclusions  A  or  I,  EA  either  E  or  O, 
AI  only  /,  EI  only  O.  We  have  then  six  moods  in  Figure  i 
which  do  not  ofiend  against  any  of  the  rules  of  the  syllogism, 
namely,  AAA,  A  A  I,  All,  EAE,  EAO,  EIO. 

We  may  establish  the  actual  validity  of  these  moods  by 
shewing  that  the  axiom  of  the  syllogism,  the  Didujn  de 
omni  ct  nullo,  applies  to  them ;  or  by  taking  them  severally 
and  shewing  that  in  each  case  the  cogency  of  the  reasonin 
is  self-evident. 

Ihe  special  rules  and  the  legitimate  moods  of  Figure  2. 
The  position  of  the  terms  in  figure  2  is  shewn  thus,— 

P-M 

S-M 

S-P 

and  its  special  rules,  (which  the  student  is  recommended  to 
deduce  from  the  general  rules  of  syllogism  for  himself), 
are, — 

(i)   O fie  premiss  7nust  be  fiegative; 

(2)   The  major  premiss  must  be  imiversaL 

Applying  these  rules,  we  shall  find  that  we  are  again  left 
with  six  moods,  namely,  AEE,  AEO,  A 00,  EAE,  EAO 
EIO. 


170 


SYLLOGISMS. 


[part  III. 


We  cannot  now  apply  the  Dictum  de  omni  d  iiullo  to 
shew  positively  that  these  moods  are  legitimate.  We  may 
however  as  before  establish  the  cogency  of  the  reasoning 
in  each  case  by  shewing  it  to  be  self-evident.  The  older 
logicians  did  not  adopt  this  course,  but  they  proved  that 
by  means  of  immediate  inferences  each  could  be  reduced 
to  such  a  form  that  the  Dictum  could  be  directly  applied 
to  it.  This  is  the  doctrine  of  Reduction  to  which  reference 
will  be  made  subsequently. 

The  special  rules  and  the  legitimate  moods  of  Figure  3. 

The  position  of  the  terms  in  this  figure  is  shewn  thus, — 

M-P 
M-S 

S-P 

and  its  special  rules  are, — 

(i)  The  minor  must  he  affirmative; 
(2)  The  conclusion  must  be  particular. 

Proceeding  as  before,  we  shall  find  ourselves  left  with 
six  valid  moods,— ^^/,  All,  EAO,  EIO,  I  A  I,  OAO. 

The  special  rules  and  the  legitimate  moods  of  Figure  4. 
The  position  of  the  terms  in  this  figure  is  shewn  thus, — 

P-M 

M-S 

S-P 

and  its  special  rules  are, — 

{i)  If  the  major  is  affirmative^  the  mitior  must  be  uni- 
versal; 

(2)  If  either  premiss  is  negative^  the  major  must  be  uni- 
versal; 


CHAP.  III.]  SYLLOGISMS.  171 

(3)  If  the  minor  is  affirmative,  the  conclusion   must  be 
particular. 

The  result  of  the  application  of  these  rules  is  again  six 
valid  moods:— ^^/,  AEE,  AFO,  FAO,  FIO,  lAL 

Our   final   conclusion    then   is  that  there  are  24    valid 
moods,  namely,  six  in  each  figure. 

In  Figure  i,  AAA,  A  A  I,  EAE,  FAO,  All,  EIO. 
In  Figure  2,  FAF,  FAO,  AFF,  AFO,  FIO,  A 00. 
In  Figure  3,  A  A  I,  I  A  I,  All,  FAO,  OAO,  FIO. 
In  Figure  4,  AAI,  AFF,  AFO,  FAO,  I  A  I,  FIO. 

145.  Weakened  Conclusions,  and  Subaltern  Moods. 

AVhen  from  premisses  that  would  have  justified  a  uni- 
versal conclusion  we  content  ourselves  with  inferring  a  par- 
ticular, (as,  for  example,  in  the  syllogism  All  M  is  P,  All  S 
is  M,  therefore.  Some  ^  is  P),  we  are  said  to  have  a  laeakened 
conclusion,  and  the  syllogism  is  said  to  be  a  weakened  syl 
logism  or  to  be  in  a  subaltern  mood,  (because  the  conclusion 
might  be  obtained  by  subaltern  opposition  from  the  con- 
clusion of  the  corresponding  strong  mood). 

In  the  preceding  section  it  has  been  shewn  that  in  each 
figure  there  are  six  moods  which  do  not  offend  against  any 
of  the  syllogistic  rules;  so  that  in  all  we  should  have  24 
distinct  valid  moods.  P^ive  of  these  however  have  weakened 
conclusions;  and,  since  we  are  not  likely  to  be  satisfied 
with  a  particular  conclusion  when  the  corresponding  uni- 
versal could  be  obtained  from  the  same  premisses,  these 
moods  are  of  no  practical  importance,  so  that  when  the  moods 
of  the  various  figures  are  enumerated  (as  in  the  mnemonic 
verses)  they  are  usually  omitted. 


172 


SYLLOGISMS. 


[part  III. 


The  subaltern  moods  are, — 
In  Figure  i,  AAI,  EAO ; 
In  Figure  2,  EAO,  AEO ; 
In  Figure  4,  AEO. 

146.  In  what  figure  can  there  be  no  weakened 
conclusion  and  why.?  Do  any  of  the  19  moods  com- 
monly recognised  give  a  weaker  conclusion  than  the 
premisses  would  warrant  t  [l.] 

It  is  obvious  that  there  can  be  no  weakened  conclusion 
in  Figure  3,  since  in  no  case  can  we  infer  more  than  a  par- 
ticular conclusion  in  this  figure. 

I  should  answer  the  question,  "whether  any  of  the  19 
moods  commonly  recognised  yield  a  weaker  conclusion 
than  the  i)remisses  would  warrant,"  in  the  negative.  Pro- 
fessor Jevons  {Studies  in  Deductive  Logic,  p.  87)  apparently 
answers  it  in  the  affirmative,  having  in  view  AAI  in  Figure  4. 

With  the  premisses 

All  P  is  M, 

All  J/ is  5; 
the  conclusion  Some  *S  is  /*  is  certainly  in  one  sense  weaker 
than  the  premisses  would  warrant  since  we  might  have 
inferred  the  universal  conclusion  All  P  is  S.  But  All  P  is  S 
is  not  the  universal  corresponding  to  Some  S  is  P,  The 
subjects  of  these  two  propositions  are  different;  and  we 
infer  all  that  we  possibly  can  about  S  when  we  say  Some 
S  is  P.  In  other  words,  regarded  as  a  mood  of  I  igure  4, 
this  mood  is  not  a  subaltern.  AAI  in  Figure  4  is  thus 
differentiated  from  AAI  in  Figure  i,  and  its  recognition  in 
the  mnemonic  verses  justified. 

I  do  not  quite  understand  Professor  Jevons's  comments 
on  this  case.     Answering  the  same  question  as  that  with 


CHAP.  III.] 


SYLLOGISMS. 


which  we  are  dealing,  he  says  '' Bmmantip'  of  the  fourth 
figure  is  the  single  mood  alluded  to  in  the  latter  part  of  the 
question.  Considering  that  it  is  impossible  to  employ  con- 
version by  limitation  without  weakening  the  logical  force  of 
the  premiss,  it  is  too  bad  of  the  Aristotelian  logicians  to 
slight  the  weakened  moods  of  the  syllogism  as  they  have 
usually  done"  {Studies,  pp.  87,  88).  The  truth  is  that  for 
practical  purposes  they  may  certainly  be  neglected";  but 
their  recognition  gives  a  completeness  to  the  theory  of  Syl- 
logism which  it  cannot  otherwise  possess.  There  is  also  a 
symmetry  in  the  result  of  their  recognition  as  yielding 
exactly  six  legitimate  moods  in  each  figure. 

147.  Strengthened  Syllogisms. 
If  in  a  syllogism,  the  same  conchision  could  be  obtained 
although  we  substituted  for  one  of  the  premisses  its  sub- 
altern, the  syllogism  is  said  to  be  a  streiigthencd  syllogism. 
A  strengthened  syllogism  is  thus  a  syllogism  with  an  un- 
necessarily strengthened  premiss. 

For  example,  the  conclusion  of  the  syllogism,—- 

AllJ/is/^, 
All  M'\%  Sy 

therefore.    Some  S  is  P, 

could  equally  be  obtained  from  the  premisses, 

AWMisP, 
Some  Mis  S; 
or  from  the  premisses, — 

Some  Mis  P, 
AUJ/is  S. 

1  i.e.,  AAI  in  Figure  4.     Cf.  section  158. 

2  A  A/ in  Figure  4  is  not  to  be  regarded  as  a  weakened  mood,  as  I 
have  just  shewn. 


174 


SYLLOGISMS. 


[part  III. 


By  trial  we  may  find  that  arry  syllogism  in  which  there 
are  two  titiivcrsal  premisses  with  a  particular  conclusion  is  a 
strengthened  syllogism,  with  the  one  exception  of  AEO  in 
the  fourth  Figured 

In  a  full  enumeration  there  are  two  strengthened  syllo- 
gisms in  each  figure  ; — 

In  Figure  i,  A  A  I,  EAO\ 

In  Figure  2,  EAO,  AEO] 

In  Figure  3,  A  A  I,  EAO  \ 

In  Figure  4,  A  A  I,  EAO. 
The  distinction  between  a  strengthened  syllogism,  (that 
is,  a  syllogism  with  a  strengthened  premiss),  and  a  weakened 
syllogism,  (that  is,  a  syllogism  with  a  weakened  conclusion), 
should  be  carefully  noted. 

It  will  be  observed  that  in  Figures  1  and  2,  a  syllogism 
having  a  strengthened  premiss  may  also  be  regarded  as  a 
syllogism  having  a  weakened  conclusion,  and  vice  versa;  but 
in  Figures  3  and  4,  the  contradictory  holds  in  both  cases. 
The  only  syllogism  with  a  weakened  conclusion  in  either  of 
these  figures  is  AEO  in  Figure  4,  but  this  does  not  contain  a 
strengthened  premiss.     That  is,  having 

All  F  is  M, 
NoMisSy 

therefore,  Some  S  is  not  F; 
the  syllogism  becomes  invalid,  if  for  either  of  the  premisses 
we  substitute  its  subaltern. 

148.  The  peculiarities  and  uses  of  each  of  the 
four  figures  of  the  syllogism. 

Figure  i.  In  this  figure  we  can  prove  conclusions  of 
all  the  forms  A,  E, /,  O ;  and  it  is  the  only  figure  in  which 

^  A  general  proof  of  this  proposition  is  given  in  section  281. 


CHAP.  III.] 


SYLLOGISMS. 


175 


we  can  prove  a  universal  afifirmative  conclusion.  This  alone 
makes  it  by  far  the  most  useful  and  important  of  the  syllo- 
gistic figures.  All  deductive  science,  the  object  of  which 
is  to  establish  universal  afiirmatives,  tends  to  work  in  AAA 
in  this  figure. 

Another  point  to  notice  is  that  only  in  this  figure  have 
we  both  the  subject  of  the  conclusion  as  subject  in  the  pre- 
misses, and  the  predicate  of  the  conclusion  as  predicate  in 
the  premisses.  (In  Figure  2  the  predicate  of  the  conclusion 
is  subject  in  the  major  premiss ;  in  Figure  3  the  subject  of 
the  conclusion  is  predicate  in  the  minor  premiss  ;  and  in 
Figure  4  we  have  a  double  inversion.)  This  is  no  doubt 
one  reason  why  reasoning  in  Figure  i  so  often  seems  more 
natural  than  the  same  reasoning  expressed  in  any  of  the 
other  figures '. 

Figure  2.  In  this  figure  we  can  prove  negatives  only ; 
and  therefore  it  is  chiefly  used  for  purposes  of  disproof. 
For  example,  Every  real  natural  poem  is  naive;  those 
poems  of  Ossian  which  Macpherson  pretended  to  discover 
are  not  naive  (but  sentimental);  hence  they  are  not  real 
natural  poems.  (Ueberweg,  System  of  Logic,  translation  by 
Lindsay,  p.  416.)  It  has  been  called  the  exclusive  figure; 
because  by  means  of  it  we  may  go  on  excluding  various 
suppositions  as  to  the  nature  of  something  under  investiga- 
tion, whose  real  character  we  wish  to  ascertain,  (a  process 
called  ahscissio  infiniti). 

For  example, 

Such  and  such  an  order  has  such  and  such  pro- 
perties. 
This  plant  has  not  those  properties  ; 
therefore,  It  does  not  belong  to  that  order. 


^  Compare  Solly,  Syllabus  of  Logic,  pp.  130 — 132, 


176 


SYLLOGISMS. 


[part  III. 


CHAP.  III.] 


SYLLOGISMS. 


^n 


This  syllogism  might  be  repeated  with  a  number  of 
dififerent  orders  till  the  enquiry  is  so  narrowed  down  that 
the  place  of  the  plant  is  easily  determined.  Whately  {Ele- 
ments of  Logic,  p.  92)  gives  an  example  from  the  diagnosis  of 
a  disease. 

/7>//r^  3.  In  this  figure  we  can  prove  particulars  only. 
It  is  frequently  useful  when  we  wish  to  take  objection  to 
a  universal  proposition  laid  down  by  an  opponent,  by- 
establishing  an  instance  in  which  such  universal  proposition 
does  not  hold  good. 

It  is  the  natural  figure  when  the  middle  term  is  a  singular 
term,  especially  if  the  other  terms  arc  general.  We  have 
already  shewn  that  if  one  and  only  one  term  of  an  affirmative 
proposition  is  singular  it  is  almost  necessarily  the  subject. 
For  example,  such  a  reasoning  as, — 
Socrates  was  wise, 
Socrates  was  a  philosopher, 

therefore,  Some  philosophers  are  wise, 

could   only   be  expressed  with  great  awkwardness  in  any 

figure  other  than  Figure  3. 

Figure  4.  This  figure  is  seldom  used,  and  some  logicians 
have  altogether  refused  to  recognise  it.  We  shall  return  to  a 
discussion  of  it  subsequently.     Compare  section  172. 

[Lambert,  (a  distinguished  mathematician  as  well  as  lo- 
gician, whose  Neues  Organon  appeared  in  1764),  expressed 
the  uses  of  the  different  syllogistic  figures  as  follows  :  *'The 
first  figure  is  suited  to  the  discovery  or  proof  of  the  pro- 
perties of  a  thing ;  the  second  to  the  discovery  or  proof  of 
the  distinctions  between  things  ;  the  third  to  the  discovery  or 
proof  of  instances  and  exceptions  ;  the  fourth  to  the  dis- 
covery or  exclusion  of  the  different  species  of  a  genus." 


,j\ 

0  *  J 
J  1 
i  ^ 

>* 


De  Morgan  {Syllabus,  p.  30)  thus  characterizes  the  dif- 
ferent figures, — 

"The  first  figure  may  be  called  the  figure  of  direct 
transition :  the  fourth,  which  is  nothing  but  the  first  with  a 
converted  conclusion\  the  figure  of  inverted  transition;  the 
second,  the  figure  of  reference  to  (the  middle  term);  the 
third,  the  figure  of  reference  from  (the  middle  term)."] 

149.  Shew  the  inadequacy  of  Hamilton's  proof  of 
the  special  rule  that  in  Figure  2  one  premiss  must  be 
negative.  "  For  were  there  two  affirmative  premisses, 
as : — 

All  Pare  J/; 
All  5  are  M -, 

All  metals  ai'e  minerals ; 
All  pebbles  are  minerals  ; 

the    conclusion    would    be — Aj.11  pebbles  are   metals^ 
which  would  be  false"  {Logic,  vol.  I,  pp.  408,  9). 

150.  Which  of  the  following  conjunctions  of  pro- 
positions make  valid  syllogisms }  In  the  case  of 
those  which  you  regard  as  invalid,  give  your  reasons 
for  so  treating  them. 

Fig.  I.  Fig.  1,  Fig.  3.  Fig.  4. 

AEE        AAA  AOE        ALL 

AGO        AOE  AEO 

LEA  A  00 

AEE  LEO 

AAL 

[c] 


^  Cf.  section  172. 


K.  L. 


12 


178  SYLLOGISMS.  [part  hi. 

151.  What  Moods  arc  good  in  the  first  figure 
and  faulty  in  the  second,  and  vice  versa  ?  Why  are 
they  excluded  in  one  figure  and  not  in  the  other  ? 

[o.] 

152.  Shew  that  O  cannot  stand  as  premiss  in 
Figure  i,  as  major  in  Figure  2,  as  minor  in  Figure  3, 
as  premiss  in  Figure  4.  [c] 

153.  Shew  that  it  is  impossible  to  have  the  con- 
clusion in  A  in  any  figure  but  the  first.  What 
fallacies  would  be  committed  if  there  were  such  a 
conclusion  to  a  reasoning  in  any  other  figure  ?    [c] 

154.  Shew  that  a  syllogism  In  Figure  4  cannot 
have  O  for  a  premiss,  nor  A  for  a  conclusion.      [c] 

155.  Prove  that  in  Figure  4,  if  the  minor  premiss 
is  negative,  both  the  premisses  must  be  universal. 


CHAPTER  IV. 


THE    REDUCTION    OF   SYLLOGISMS. 


156.  The  Problem  of  Reduction, 

By  Reduction  is  meant  the  process  of  expressing  the 
reasoning  contained  in  a  given  syllogism  in  some  other 
mood  or  figure.  Unless  otherwise  stated,  Reduction  is 
always  supposed  to  be  to  Figure  i. 

As  an  example,  we  may  take  the  following  syllogism  in 
Figure  3,— 

AllJ/Is/', 

Some  M  is  5, 

therefore,  Some  S\%  P, 

It  will  be  seen  that  by  simply  converting  the  minor  pre- 
miss, we  have  precisely  the  same  reasoning  in  Figure  i. 
This  is  an  example  of  direct  or  ostensive  reduction. 

157.  Indirect  Reduction. 

We  prove  a  proposition  indirectly  when  we  prove  its 
contradictory  to  be  false  ;  and  this  may  be  done  by  shewing 
that  an  ultimate  consequence  of  tlie  truth  of  its  contra- 
dictory is  the  truth  of  some  proposition  that  is  self-evidently 
false. 

The  method  of  indirect  proof  is  in  several  cases  adopted 
by  Euclid ;  and  it  is  sometimes  employed  in  the  reduction 

12 — 2 


i8o  SYLLOGISMS.  [part  hi. 

of  syllogisms  from  one  mood  to  another.  Thus,  AOO  in 
Figure  2  is  usually  reduced  in  this  manner.  From  the 
premisses, — 

All  P  is  M, 
Some  5  is  not  J/, 

it  follows  that  Some  6*  is  not  P  ] 

for  if  this  conclusion  is  not  true,  its  contradictory  (namely, 
All  S  is  P),  must  be  so,  and  the  premisses  being  given  true 
we  shall  have  true  together  the  three  propositions, — 

All  Pis  J/;    (i) 

Some  6*  is  not  M;  (2) 

KWS'isP.      (3) 

But  combining  (i)  and  (3)  we  have  a  syllogism  in 
Figure  i, — 

All  P  is  M, 
All  S  is  P, 

yielding  the  conclusion  All  S  is  M.     (4) 

Some  5  is  not  M  (2),  and  All  »S  is  M  (4)  are  therefore 
true  together;  but  this  is  self-evidently  absurd,  since  they 
are  contradictories. 

Hence  it  has  been  shewn  that  the  consequence  of  sup- 
posing Some  6*  is  not  P  false  is  a  self-contradiction ;  and 
we  may  therefore  infer  that  it  is  true. 

It  will  be  observed  that  the  only  explicit  syllogism  that 
has  been  made  use  of  in  the  above  is  in  Figure  i ' ;  and  the 

^  Solly  {Syl/alfiis  of  Logic,  p.  104)  maintains  that  a  full  analysis  of 
the  reasoning  will  shew  that  three  distinct  syllogisms  are  really  in- 
volved,— 

*'  Let  A  and  B  represent  the  premisses,  and  C  the  conclusion  of  any 
syllogism.  In  order  to  prove  C  by  the  indirect  method,  we  commence 
with  assuming  that  C  is  not  true.  The  three  syllogisms  may  be  then 
.stated  as  follows : 


CHAP.  IV.] 


SYLLOGISMS. 


i8f 


I)rocess  is  therefore  regarded  as  a  reduction  of  the  reasoning 
to  Figure  i. 

This  method  of  reduction  is  called  Redudio  ad  impossibile, 
or  Redudio  per  impossibile\  ox  Deductioad  impossibile,  ox  De- 
dudio  ad  absurd nm.  It  is  the  only  way  of  reducing  AOO 
(Figure  2),  cr  OAO  (Figure  3),  to  Figure  i,  unless  we  make 
use  of  negative  terms  (as  in  obversion  and  contraposition) ; 
and  it  was  adopted  by  the  old  writers  in  consequence  of 
their  objection  to  negative  terms, 

158.  The  mnemonic  lines  Barbara,  Celarent,  &ic. 
The  mnemonic  verses,  (which  are  spoken  of  by  De  Mor- 
gan as  ''the  magic  words  by  which  the  different  moods 
have  been  denoted  for  many  centuries,  words  which  I  take 
to  be  more  full  of  meaning  than  any  that  ever  were  made  "), 
are  usually  given  as  follows, — 

Barbara  J   Celarent,  Darii^  PcrioquQ  prioris: 
Cesare,   Camcstres,  Festiiio,  Baroco,  secundae : 
Tertia,  Darapti,  Disainis,  Datisi,  Felaptou, 
Bocardoy  Ferison,  habet :    Quarta  insuper  addit 
Bramautip,    Camcues^  Dimaris^   Fcsapo,  Fresison. 

Each  valid  mood  in  every  figure,  unless  it  be  a  subaltern 

First  syllogism  :  M  is  ;  C  is  not ;  therefore  B  is  not'. 

Second  sylloj;ism :  '  If  A  is,  and  C  is  not,  it  follows  that  B  is  not ; 
but  B  is;  therefore  it  is  false  that  A  is  and  C  is  not." 

Third  syllogism :  '  Either  both  propositions  ./  is  and  C  is  not  are 
false,  or  else  one  of  them  is  false;  but  that  A  is  is  not  false;  therefore 
that  C  is  not  is  false,  (/. ^.,  C  is).'" 

I  do  not  see  any  flaw  in  this  analysis;  at  any  rate  it  must  be 
admitted  that  the  reasoning  involved  in  Indirect  Reduction  is  highly 
complex,  and  since  the  two  moods  to  which  it  is  generally  applied  can 
also  be  reduced  directly  (compare  section  159),  some  modern  logicians 
are  inclined  to  banish  it  entirely  from  their  treatment  of  the  syllogism. 

*  Cf.  Mansel's  Aldrich,  pp.  88,  89. 


l82 


SYLLOGISMS. 


[part  III. 


mood,  is  here  represented  by  a  separate  word ;  and  in  the 
case  of  a  mood  in  any  of  the  so-called  imperfect  figures, 
{i.e.,  Figures  2,  3,  4),  the  mnemonic  gives  full  information 
for  its  reduction  to  Figure  i,  the  so-called  perfect  figure. 

The  only  meaningless  letters  are  b  (not  initial),  //,  /, ;/,  ;*, 
/  ;  the  signification  of  the  remainder  is  as  follows  : — 

T/ie  vowels  give  the  quality  and  quantity  of  the  propo- 
sitions of  which  the  syllogism  is  composed;  and  therefore' 
really   give  the  syllogism  itself.     Thus,   Camencs  being  in 
Figure  4,  represents  the  syllogism, — • 

All  P  is  My 
No  M  is  S, 
therefore,  No  S  is  P, 

The  initial  letters  in  the  case  of  Figures  2,  3,  4  shew  to 
which  of  the  moods  of  Figure  i  the  given  mood  is  to  be 
reduced,  namely  to  that  which  has  the  same  initial  letter. 
[The  letters  B,  Q  Z>,  F  were  chosen  for  the  moods  of  Figure  i 
as  being  the  first  four  consonants  in  the  alphabet.] 

Thus,  Camestres  is  reduced  to  Celarent, — 


All  P  is  M, 
No  S  is  J/, 


X 


No  M  is  5, 
All  P  is  M, 


therefore,        No  S  is  P.  therefore,  No  P  is  Sy 

therefore.  No  S  is  P. 

s  (in  the  middle  of  a  word)  indicates  that  in  the  process 
of  reduction  the  preceding  proposition  is  to  be  simply  con- 
verted. Thus,  in  reducing  Camestres  to  Celare?ity  as  shewn 
above,  the  minor  premiss  is  simply  converted. 

s  (at  the  end  of  a  word')  shews  that  the  conclusion  of 
the  ne7i'  syllogism  has  to  be  simply  converted  in  order  to 


^  This  slight  difference  in  the  signification  of  J  and  /  when  they  are 
final  letters  is  frequently  overlooked. 


CHAP.  IV.] 


SYLLOGISMS. 


183 


obtain  the  given  conclusion.  This  again  is  illustrated  in  the 
reduction  of  Camestres.  The  final  s  does  not  affect  the  con- 
clusion of  Camestres  itself,  but  the  conclusion  of  Celarent  to 
which  it  is  reduced. 

/  (in  the  middle  of  a  word)  signifies  that  the  preceding 
proposition  is  to  be  converted  per  aceidens.  Thus,  in  the 
reduction  of  Daraptl  to  Darii, — 

AW  AfhP,  AWM'isPy 

All  M  is  S,  Some  S  is  J/, 

therefore.      Some  S  is  P.    therefore.  Some  S  is  P. 

p  (at  the  end  of  a  word')  implies  that  the  conclusion 
obtained  by  reduction  is  to  be  converted  per  accide?ts.  Thus, 
in  Bramantipy  the  p  obviously  cannot  affect  the  /  conclusion 
of  the  mood  itself;  it  really  affects  the  A  conclusion  of  the 
syllogism  in  Barbara  which  is  given  by  reduction.  Thus, — 
All  P  is  M,  .^  All  M  is  S, 
All  AI  is  S,        ^       All  P  is  Af, 


therefore,       Some  S  is  P.  therefore,  All  P  is  Sy 

therefore,  Some  S  is  P, 

m  indicates  that  in  reduction  the  premisses  have  to  be 
transposed,  {Metathesis prcemissariim) ;  as  just  shewn  in  the 
case  of  Brama7itip. 

c  signifies  that  the  mood  is  to  be  reduced  indirectly y  {i.e., 
by  reductio  per  impossibile  in  the  manner  indicated  in  the 
preceding  section) ;  and  the  position  of  the  letter  indicates 
that  in  this  process  of  indirect  reduction  the  first  step  is  to 
omit  the  premiss  preceding  it,  i.e.,  the  other  premiss  is  to  be 
combined  with  the  contradictory  of  the  conclusion,  {Con- 
versio  syllogis?ni,  or  ductio  per  Contradictoriam  propositioiiem 
sive  per  impossibile).  c  is  by  some  writers  replaced  by  >(',  thus 
Baroko  and  Bokardo  instead  of  Baroco  and  Bocardo, 

^  See  note  on  the  preceding  page. 


i84  SYLLOGISMS.  [part  hi. 

The  following  lines  are  sometimes  added  to  the  verses 
given  above,  in  order  to  meet  the  case  of  the  subaltern 
moods; — 

Quinque  Subaltcrni,  totidem  Generalibus  orti, 
Nomen  habent  nullum,  nee,  si  bene  colligis.  usum\ 

159.  The  direct  reduction  of  Baroco  and  Bocardo, 
Mnemonics  representing  the  direct  reduction  of  these 
moods. 

^  Tlic  mnemonics  have  been  written  in  various  forms.  Those  given 
above  are  from  Aklrich,  and  they  ^re  the  ones  that  are  in  general  use 
in  England.  Wallis  in  his  Institiitio  Logiccp.  (1687)  gives  for  Figure  4, 
Balani^  Cfldere^  Digami^  Fcgano^  Fedibo,  P.  van  Musschenbroek  in 
his  Instiititioiics  Logiae  (1748)  gives  Barbaric  Calentes^  Dibalis,  Fcs- 
paniOy  FrcsisoJH.  1  his  variety  of  forms  for  the  moods  of  Figure  4  wns 
no  doubt  due  to  the  fact  that  the  recognition  of  this  figure  at  all  was 
quite  exceptional  until  comparatively  recently.     Compare  section  173. 

According  to  Ueberweg,  the  mnemonics  run, — 

Barbara^   Celarcut  prima;,  Darii  /Vr/^qup. 
Cesarc^  Camestres,  Fcstino^  Baroco  secundx. 
Terlia  grande  sonans  recitat  Darapti^  Fclapton^ 
DisamiSy  £)atisi,  Bocardo,  Fcrison.     Quartae 
Sunt  Bamalip^   Calcines^  Dimatis,  Fcsapo^  Frcsison. 

Mr  Carveth  Read  [Mind,  18S2,  p.  440)  suggests  an  ingenious 
jnodification  of  the  verses,  so  as  to  make  each  mnemonic  immediately 
suggest  the  figure  to  which  the  mood  it  represents  belongs,  at  the 
same  time  abolishing  all  the  unmeaning  letters.  lie  takes  /  as  the  sign 
of  the  first  figure,  11  of  the  second,  r  of  the  third,  and  i  of  the  fourth. 
The  lines  then  run 

Ballala,   Celallel,  Dalit,  Fclio<\\xc  prioris. 
Cesaney   Games nes,  Fcsinon,  Banoco  secundoe. 
Tertia  Darapri^  Drisatuis,  Darisiy  Ferapro^ 
Bocaro,  Fcrisor  habet.     Quarta  insuper  addit 
BamatiPy   Cametes,  Dimatisy  Fcsapto,  Fcsistot. 

Mr  Read  also  suggests  mnemonics  to  indicate  the  direct  reduction 
of  Baroco  and  Bocardo.     Compare  the  following  section. 


CHAP.  IV.] 
Bai'oco : — 


SYLLOGISMS. 

All  P  is  M, 

Some  S  is  not  M^ 


185 


therefore,  Some  S  is  not  P^ 

may  be  reduced  to  Fefio  by  contrapositing  the  major  pre 
miss,  and  obverting  the  minor  premiss,  thus, — 

No  not-J/  is  P, 
Some  S  is  not-J/, 

therefore,  Some  S  is  not  P. 

Professor  Groom  Robertson  has  suggested  Faksoko  to 
represent  this  method  of  reduction,  k  denoting  obversion,  so 
"^      that  ks  denotes  obversion  followed  by  conversion,  {i.e.,  con- 
traposition). 

Whately's  word  Fakoj'o  {Elements  of  LogiCy  p.  97)  does 
not  indicate  the  obversion  of  the  minor  premiss  (r  being 
with  him  an  unmeaning  letter). 

Bocardo  : — 

Some  M  is  not  /*, 

All  M  is  5, 

therefore,  Some  5  is  not  P, 

may   be   reduced    to   Darii    by   contrapositing   the   major 
premiss  and  transposing  the  premisses,  thus, 

All  M  is  S, 
Some  not-/*  is  M^ 

therefore,  Some  not-/' is  S. 

We  have  first  to  convert  and  then  to  obvert  this  conclu- 
sion, however,  in  order  to  get  the  original  conclusion.  This 
process  may  be  indicated  by  Doksaniosk,  (which  again  is 
obviously   preferable    to    Dokamo   suggested    by   Whately, 


1 86 


SYLLOGISMS. 


[part  III. 


since  this  word  would  make  it  appear  as  if  we  immediately 
obtained  the  original  conclusion  in  Darii^). 

160.  Shew  how  to  reduce  Bramantip  by  the 
indirect  method. 

Just  as  Bocardo  and  Baroco  which  arc  usually  reduced 
indirectly  may  be  reduced  directly,  so  other  moods  which 
are  usually  reduced  directly  may  be  reduced  indirectly. 

Bramantip : — 

All  P  is  M, 
All  J/ is  S, 
therefore,  Some  »S  is  7^ ; 
for,  if  not,   then  No  S  is  P\  and  combining  this  with  the 
given  minor  premiss  we  have  a  syllogism  in  Cdarcnt^ — 

No  S  is  P, 

All  M  is  5, 
therefore,  No  J/ is  P, 
which  yields  by  conversion  No  P  is  M.     But  this  is  the 
contrary  of  the  original  major  premiss  All  P  is  M,  and  it  is 
impossible  that  they  should  be  true  together.     Hence  we 
infer  the  truth  of  the  original  conclusion. 

161.  Assuming  that  any  syllogistic  reasoning  can 
be  expressed  in  the  first  Figure,  prove  that,  (omitting 
the  subaltern  moods),  it  can  be  expressed,  directly  or 
indirectly,  in  any  given  mood  of  that  Figure. 

^  Mr  Carveth  Read  {Mind,  1882,  p.  441)  uses  the  letters  k  and  s  as 
above;  but  his  mnemonics  are  required  also  to  indicate  the  figure  to 
which  the  moods  belong  (compare  the  preceding  note) ;  and  he  there- 
fore arrives  at  Faksnoko  and  Doksamrosk. 

Spalding  (Z^,?-/V,  p.  235)  suggests  Facoco  and  Docamoc\  but  the 
processes  here  indicated  by  the  letter  c  are  not  in  all  cases  the  same, 
and  these  mnemonics  are  therefore  unsatisfactory. 


CHAP.  IV.] 


SYLLOGISMS. 


187 


We  may  extend  the  doctrine  of  reduction,  and  shew  not 
merely  that  any  syllogism  may  be  reduced  to  Figure  i,  but 
also  that  it  may  be  reduced  to  any  given  mood  of  that 
figure,  provided  it  is  not  a  subaltern  mood.  This  position 
will  obviously  be  established  if  we  can  shew  that  Barbara, 
Celarciit,  Darii  and  Fcrio  are  mutually  reducible  to  one 
another.  Barbara  may  be  reduced  to  Cdarent  by  obverting 
the  major  premiss  and  also  the  new  conclusion  which  is 
thereby  obtained.     Thus, 

All  M  is  P, 
All  S  is  M, 


therefore,  All  5  is  P, 
becomes  No  M  is  not-/', 
All  ^  is  M, 


therefore.  No  6*  is  not-/*, 
therefore.  All  S  is  P. 

Conversely,  Cclarait  is  reducible  to  Barbai'a ;  and  in  a 
similar  manner  by  obversion  of  major  premiss  and  con- 
clusion Darii  and  Fcrio  are  reducible  to  each  other. 

It  will  now  suffice  if  we  can  shew  that  Barbara  and 
Darii  are  mutually  reducible  to  each  other.  Obviously  the 
only  method  possible  here  is  the  uidircct  method. 


Take  Barbara, 


MaP, 
SaM, 

SaP; 


for,  if  not,  then  we  have  SoP;  and  MaP,  SaM,  SoP  must 
be  true  together.  From  SoP  by  first  obverting  and  then 
converting,  (and  denoting  not-/*  by  P'),  we  get  P'iS,  and 
combining  this  with  SaM  we  have  a  syllogism  in  Darii, — 


i88 


SYLLOGISMS. 


[part  hi. 


P'iM, 

P'iM  by  conversion  and  obversion  becomes  MoP\  and 
therefore  MaP  and  MoP  are  true  together ;  but  this  is  im- 
possible, since  they  are  contradictories. 

Therefore,  SoP  cannot  be  true,  i.e.,  the  truth  of  SaP  is 
estabHshed. 

Similarly,  Da rii  m^y  be  indirectly  reduced  to  Barbara^ 

MaP,     (i) 
SiJ\f,      (ii) 

SiP.      (iii) 

The  contradictory  of  (iii)  is  SeP,  from  which  we  obtain  PaS'. 
Combining  with  (i),  we  have — 

PaS' 
MaP, 

MaS'  in  Barbara. 

But  from  this  conclusion  we  may  obtain  SeM,  which  is  the 
contradictory  of  (ii)^ 

162.  Some  logicians  have  asserted  that  all  the 
moods  of  the  syllogism  arc  reducible  to  the  form  of 
Barbara.    Inquire  into  the  truth  of  this  assertion,    [l.] 

163.  Making  use  of  any  legitimate  methods  of 
immediate   inference  that   may   be   serviceable,  shew 

^  It  has  also  been  maintained,  that  this  reduction  is  unnecessary, 
and  that,  to  all  intents  and  purposes,  Dari't  is  Barbara,  since  the 
*'  some  ^S'"  in  the  minor  is,  and  is  known  to  be,  the  same  some  as  in  the 
conclusion. 

2  It  would  now  seem  that  the  Dictum  de  omni  ct  ntiUo  might  if  we 
pleased  be  reduced  to  a  Dictum  de  omni ;  but  it  would  be  vain  to  pre- 
tend that  any  real  simplification  would  be  introduced  thereby. 


CHAP.  IV.]  SYLLOGISMS.  189 

how  Barbara,  Baroco  and   Bccardo  may  be  reduced 
ostensively  to  Figure  4. 

164.  Reduce  Fcrio  to  Figure  2,  Festijw  to  Figure 
3,  F Clapton  to  Figure  4. 

165.  Prove  that  any  mood  may  be  reduced  to 
any  other  mood  provided  that  the  latter  contains 
neither  a  strengthened  premiss  nor  a  weakened  con- 
clusion. 

166.  Examine  the  following  statement  of  De 
Morgan's  :—'^  There  arc  but  six  distinct  syllogisms. 
All  others  are  made  from  them  by  strengthening 
one  of  the  premisses,  or  converting  one  or  both  of 
the  premisses,  where  such  conversion  is  allowable; 
or  else  by  first  making  the  conversion,  and  then 
strengthening  one  of  the  premisses." 

167.  How  can  you  apply  the  Dictum  de  omni  et 
nnllo  to  the  following  syllogism  : — Some  M  is  not  P, 
All  y]/is  S,  therefore,  Some  6"  is  not  P  ? 

168.  How  would  you  apply  the  Dictum  de  omni 
ct  nulla  to  the  following  reasonings  t 

(i)  The  life  of  St  Paul  proves  the  falsity  of  the 
conclusion  that  only  the  rich  are  happy. 

(2)  His  weakness  might  have  been  foretold  from 
his  proneness  to  favourites,  for  all  weak  princes  have 
that  failing.  ryi 

169.  Dicta  for  the  second  and  third  Figures  of 
syllogism  corresponding  to  the  Dictum  of  the  first. 

Thomson  {La7vs  of  Thought,  p.  173),  and  Bowen  {Logic, 
p.   196),  give  for  Figure  2,  a  dictum  de  diverso,—''li  one 


IQO 


SYLLOGISMS. 


[part  III. 


term  is  contained  in,  and  another  excluded  from,  a  third 
term,  they  are  mutually  excluded";  and  for  Figure  3,  a 
Dictum  de  cxempio, — '*Two  terms  which  contain  a  common 
part,  partly  agree,  or  if  one  one  contains  a  part  which 
the  other  does  not,  they  partly  differ."  The  former  of 
these  is  at  least  expressed  loosely  since  it  would  appear 
to  warrant  a  universal  conclusion  in  Fcstiiio  and  Baroco, 
Mansel  {Aldrich,  p.  86)  puts  this  Dictum  in  a  more  satis- 
factory form: — **  If  a  certain  attribute  can  be  predicated, 
affirmatively  or  negatively,  of  every  member  of  a  class,  any 
subject  of  which  it  cannot  be  so  predicated,  does  not  belong 
to  the  class."  This  proposition  may  claim  to  be  axiomatic, 
and  it  can  be  applied  directly  to  any  syllogism  in  P^igure  2. 

The  Dictum  de  cxcmplo  again  as  stated  above  is  open  to 
exception.  The  proposition,  *' If  one  term  contains  a  part 
which  another  does  not  they  partly  differ,"  applied  to  No  M 
is  jP,  All  M  is  »S,  would  appear  to  justify  Some  P  is  not  S 
just  as  much  as  Some  S  is  not  P.  Mansel's  amendment 
here  is  to  give  two  principles  for  Figure  3,  the  Dictum  de 
exemplo, — **If  a  certain  attribute  can  be  affirmed  of  any 
portion  of  the  members  of  a  class,  it  is  not  incompatible 
with  the  distinctive  attributes  of  that  class" ;  and  the  Dictum  de 
excepto^ — "  If  a  certain  attribute  can  be  denied  of  any  portion 
of  the  members  of  a  class,  it  is  not  inseparable  from  the 
distinctive  attributes  of  that  class."  But  is  it  essential  that 
in  the  minor  premiss  we  should  be  predicating  the  distinctive 
attributes  of  the  class  as  is  here  implied  ?  This  appears  to 
be  a  fatal  objection  to  Mansel's  dicta  for  Figure  3.  More- 
over, granted  that  P  is  7iot  incompatible  with  5,  are  we  there- 
fore justified  in  saying  Some  S  is  /*? 

I  would  suggest  the  following  axioms, — "  If  two  terms 
are  both  affirmatively  predicated  of  a  common  third,  and 
one  at   least  of  them   universally   so,   they  may  be  par- 


CHAP.  IV.] 


SYLLOGISMS. 


191 


tially  predicated  of  each  other " ;  "  If  one  term  is  denied 
while  another  is  affirmed  of  a  common  third  term,  either 
the  denial  or  the  affirmation  being  universal,  the  former 
may  be  partially  denied  of  the  latter."  These  will  I  think 
be  found  to  apply  respectively  to  the  affirmative  and 
negative  moods  of  Figure  3,  and  they  may  be  regarded 
as  axiomatic  ;  but  they  are  certainly  somewhat  laboured. 

170.    Is  Reduction  an  essential  part  of  the  doctrine 
of  the  syllogism.'^ 

According  to  the  original  theory  of  Reduction,  the  object 
of  the  process  was  to  be  sure  that  the  conclusion  was  a  valid 
inference  from  the  premisses.  Given  a  syllogism  in  Figure  i, 
we  are  able  to  test  its  validity  by  reference  to  the  Dictum  de 
omni  et  nullo;  but  we  have  no  such  means  of  dealing  directly 
with  syllogisms  in  any  other  figure.  Thus,  Whately  says, — 
"As  it  is  on  the  Dictum  de  omni  et  nullo  that  all  Reasoning 
ultimately  depends,  so,  all  arguments  may  be  in  one  way  or 
other  brought  into  some  one  of  the  four  Moods  in  the  First 
Figure :  and  a  Syllogism  is,  in  that  case,  said  to  be  reduced'^ 
{Elemefits  0/  Logic,  p.  93).  Professor  Fowler  puts  the  same 
position  in  a  more  guarded  manner, — '*As  we  have  adopted 
no  canon  for  the  2nd,  3rd,  and  4th  figures,  we  have  as  yet 
no  positive  proof  that  the  six  moods  remaining  in  each  of 
those  figures  are  valid;  we  merely  know  that  they  do  not 
offend  against  any  of  the  syllogistic  rules.  But  if  we  can 
reduce  them,  i.e.,  bring  them  back  to  the  ist  figure,  by  shew- 
ing that  they  are  only  different  statements  of  its  moods,  or 
in  other  words,  that  precisely  the  same  conclusions  can  be 
obtained  from  equivalent  premisses  in  the  ist  figure,  their 
validity  will  be  proved  beyond  question"  (Deductive  Logic, 

p.  97). 

On  the  other  hand,  by  some  logicians    Reduction  is 


192 


SYLLOGISMS. 


[part  III. 


regarded  as  unnecessary  and  tmnaturoL  It  Is  maintained  to 
be  unnecessary  on  the  ground  that  it  is  not  true  that  the 
Dictum  de  omni  et  nulla  is  the  paramount  Liw  for  all  i)erfect 
inference,  or  that  the  first  figure  is  alone  perfect ^  In  the 
preceding  section  we  have  discussed  dicta  for  the  other 
figures,  which  may  be  regarded  as  making  them  independent 
of  the  first,  and  putting  them  on  a  level  with  it.  It  may 
also  be  maintained  that  in  any  mood  the  validity  of  a  par- 
ticular syllogism  is  as  self-evident  as  that  of  the  Dictum 
itself;  and  that  therefore  although  axioms  of  syllogism  are 
useful  as  generalisations  of  the  syllogistic  process,  they  are 
needless  in  order  to  establish  the  validity  of  any  given  syllo- 
gism.    This  view  is  indicated  by  Ueberwcg. 

Again,  Reduction  is  said  to  be  unnatural^  inasmuch  as 
it  often  involves  the  substitution  of  an  unnatural  and  indirect 
for  a  natural  and  direct  predication.  Figures  2  and  3  at 
any  rate  have  their  special  uses,  and  certain  reasonings 
naturally  fall  into  these  figures  rather  than  into  Figure  i. 
This  argument  is  very  well  elaborated  by  Archbishop 
Thomson  {Laws  of  Thought^  pp.  173 — 17*5).  He  gives  this 
example, — "Thus,  when  it  was  desirable  to  shew  by  an 
example  that  zeal  and  activity  did  not  always  proceed  from 
selfish  motives,  the  natural  course  would  be  some  such  syl- 
logism as  the  following.  The  Apostles  so^ight  no  earthly 
reward,  the  Apostles  were  zealous  in  their  work;  therefore, 
some  zealous  persons  seek  not  earthly  reward."  In  reducing 
this  syllogism  to  Figure  i,  we  have  to  convert  our  minor  into 
"Some  zealous  persons  were  Apostles,"  which  is  awkward 
and  unnatural. 

Take  again  this  syllogism, 

"  Every  reasonable  man  wishes  the  Reform  Bill  to  pass. 


^  Cf.  Thomson,  Laws  of  Thought ^  p.  172. 


CHAP.  IV.] 


SYLLOGISMS. 


^93 


I  don't, 
therefore,  I  am  not  a  reasonable  man." 

^  Reduced  in  the  regular  way  to  Celarent,  the  major  pre- 
miss becomes  "No  person  wishing  the  Reform  Bill  to  pass 
is  I,"  yielding  the  conclusion,  "No  reasonable  man  is  L" 

Further  illustrations  of  this  point  will  be  found  if  we 
reduce  to  Figure  i,  syllogisms  with  such  premisses  as  the 
following : — 

JBashfulness  Is  not  praiseworthy, 
(Modesty  is  praiseworthy. 

rSocrates  is  poor, 
(Socrates  is  wise. 

The  above  arguments  appear  conclusively  to  establish 
the  view  that  Reduction  is  not  an  essential  part  of  the  doc- 
trine of  Syllogism,  at  any  rate  so  far  as  establishing  the 
validity  of  the  different  moods  is  concerned. 

It  may,  however,  be  doubted  whether  any  treatment  of 
the  Syllogism  can  be  regarded  as  scientific  or  complete  until 
the  equivalence  between  the  moods  in  the  different  figures 
has  been  shewn;  and  for  this  purpose,  as  well  as  for  its 
utility  as  a  logical  exercise,  a  full  treatment  of  the  problem 
of  Reduction  should  be  retained. 

171.  Discuss  Hamilton's  doctrine  that  Figures  2, 
3,  and  4,  are  not  genuine  and  original  forms  of  reason- 
ing. 

"The  last  three  figures,"  says  Hamilton  {Logic,  i.  p.  433), 
"are  virtually  identical  with  the  first."  This  has  been  recog- 
nised by  logicians,  and  hence  "the  tedious  and  disgusting 
rules  of  their  reduction."  He  himself  however  goes  further, 
and  extinguishes  these  figures  altogether,  as  being  merely 


K.  L. 


13 


194 


SYLLOGISMS. 


[part  III. 


"accidental  modifications  of  the  first,"  and  "the  mutilated 
expressions  of  a  complex  mental  process." 

If  the  last  three  figures  are  admitted  as  genuine  and 
original  forms  of  reasoning,  the  following  anomalies  in 
Hamilton's  opinion  result : — 

"In  the  first  place,  the  principle  that  all  reasoning  is  the 
recognition  of  the  relation  of  a  least  part  to  a  greatest  whole, 
through  a  lesser  whole  or  greater  part,  is  invalidated.'*  In 
reply  to  this,  it  may  simply  be  asked  whether  it  really 
requires  the  last  three  figures  to  invalidate  this  principle. 

"In  the  second  place,  the  second  general  rule  I  gave 
you  for  categorial  syllogisms,  is  invalidated  in  both  its 
clauses."  It  does  not  occur  to  Hamilton  that  his  rules  may 
have  been  needlessly  limited  in  their  application.  The 
fact  is  that  he  has  with  a  great  flourish  of  trumpets  simpli- 
fied the  rules  of  the  syllogism  by  replacing  those  usually 
given  by  the  special  rules  of  Figure  i  ^ ;  and  he  is  now  shocked 
to  find  that  these  do  not  apply  to  Figures  2,  3,  4.  This 
whole  reasoning  of  Hamilton's  is  a  flagrant  example  of 
petitio  principii. 

The  question  at  issue  is  really  this, — can  we  formulate  a 
principle  which  shall  be  accepted  as  axiomatic,  and  which 
shall  apply  immediately  to  syllogisms  in  other  figures  than 
the  first? 


^  "  Had  Dr  Whately  looked  a  little  closer  into  the  matter,  he  might 
have  seen  that  the  six  rules  which  he  and  other  logicians  enumerate, 
may,  without  any  sacrifice  of  precision,  and  with  even  an  increase  of 

perspicuity,  be  reduced  to  three These  three  simple  laws  comprise 

all  the  rules  which  logicians  lay  down  with  so  confusing  a  minuteness  " 
(Hamilton,  Logic,  I.  pp.  305,  6).  But  as  I  have  remarked  in  the  text, 
the  simplification  is  obtained  solely  by  giving  laws  which  have  a  more 
limited  application  than  other  logicians  had  contemplated. 


CHAP.  lY.]  SYLLOGISMS.  195 

Now  take  a  syllogism  in  Ccsare, — 

No  P  is  J/, 
All  5  is  J/; 
therefore,  No  S  is  P, 

Hamilton  maintains  {Logic,  i.  pp.  434,  435)  that  we  can 
only  properly  see  the  force  of  this  reasoning  by  mentally 
converting  the  major  premiss  to  No  ^is  P.  But  will  not 
the  following  which  applies  immediately  to  Cesare  be  accepted 
as  axiomatic,— "If  one  class  is  excluded  from  and  another 
is  contained  in  a  third  class,  the  second  class  is  excluded 
from  the  first".^  This  simple  case  seems  to  me  sufficient  to 
overthrow  the  whole  of  Hamilton's  elaborate  but  confused 
reasoning  \ 

Perhaps  Baroco  is  a  still  better  case, — 

All  P  is  M, 
Some  6"  is  not  M, 
therefore.     Some  S  is  not  P, 

Axiom:  "If  one  class  is  totally  contained  in,  and  an- 
other partially  excluded  from  a  third  class,  the  second  class 
is  partially  excluded  from  the  first."  Now  compare  Hamil- 
ton's elaborate  explication  {Logic,  i.  pp.  438,  9), — "  The 
formula  of  Baroco  is  : — 

All  P  are  M-, 
But  some  S  are  not  M-, 
Therefore,  some  S  are  not  P, 

But  the  following  is  the  full  mental  process  : — 

Sumption, All /^  are  J/; 

Real  Subsumption, (Some  not- J/ are  S)\ 

^  It  may  be  pointed  out  that  Hamilton  himself  elsewhere  {Logic,  11. 
P-  358)  gives  special  Canons  for  Figures  2,  3. 

13—2 


196 


SYLLOGISMS. 


[part  III. 


which  gives  the 

_.  ,   f.  ,  •  fThen,  Some  S  are  not- J/: 

Expressed   Siibsumption, ...    i^     ^  ^  ,, 

(Or,  Some  o  are  not  M; 

Real  Conclusion, (Therefore,  Some  not-P  are  S) ; 

which  gives  the 

T.  J   ^       1     •  (Then,  Some  5 are  not-P: 

Expressed  Conclusion, "^^^ 

^  (Or,  Some  S  are  not  PJ' 

It  is  surely  absurd  to  say  that  we  go  through  tliis  com- 
plex mental  process  in  order  to  discover  the  validity  of  a 
syllogism  in  Baroco, 

But  even  granting  that  this  is  the  case,  I  cannot  see  how 
on  his  own  grounds  Hamilton  succeeds  in  getting  rid  of  the 
necessity  of  "the  tedious  and  disgusting"  rules  of  reduction; 
nor  that  he  has  advanced  beyond  the  logicians  who  reject 
the  independent  validity  of  Figures  2,  3,  4,  and  consequently 
establish  the  necessity  of  the  process  of  reduction,  and 
naturally  along  with  it  of  rules  for  conducting  the  process. 
It  would  seem  that  only  those  logicians  who,  like  Thomson, 
maintain  the  independent  validity  of  other  figures  than  the 
first  have  any  justification  whatever  for  ignoring  the  doctrine 
of  reduction. 

172.     The  Fourth  Figure. 

Figure  4  was  not  as  such  recognised  by  Aristotle ;  and 
its  introduction  having  been  attributed  by  Averroes  to 
Galen,  it  is  frequently  spoken  of  as  the  Galcnian  figure. 
It  does  not  usually  appear  in  works  on  logic  before  the 
beginning  of  the  last  century,  and  even  by  modern  logicians 
its  use  is  sometimes  condemned.  Thus,  Bowen  {Logic,  p. 
192)  holds  that  "what  is  called  the  Fourth  Figure  is 
only  the  First  with  a  converted  conclusion ;  that  is,  we  do 
not  actually  reason  in  the  Fourth,  but  only  in  the  First, 
and  then  if  occasion  requires,  convert  the  conclusion  of  the 


CHAP.  IV.]  SYLLOGISMS.  197 

First."  But  unless  we  quantify  the  predicate  this  account 
of  the  Fourth  Figure  cannot  be  accepted,  since  it  will  not 
apply  to  Fesapo  or  Fresisofi,  For  example,  the  premisses  of 
Fesapo  are, — 

No /'is  J/, 

AllJ/is  S\ 

and,  as  they  stand,  we  cannot  obtain  any  conclusion  whatever 
from  them  in  Figure  i. 

Thomson's  ground  of  rejection  is  that  *'in  the  fourth 
figure  the  order  of  thought  is  wholly  inverted,  the  subject  of 
the  conclusion  had  only  been  a  predicate,  whilst  the  pre- 
dicate had  been  the  leading  subject  in  the  premiss.  Against 
this  the  mind  rebels ;  and  we  can  ascertain  that  the  conclu- 
sion is  only  the  converse  of  the  real  one,  by  proposing  to 
ourselves  similar  sets  of  premisses,  to  which  we  shall  always 
find  ourselves  supplying  a  conclusion  so  arranged  that  the 
syllogism  is  the  first  figure,  with  the  second  premiss  first'* 
{Laivs  of  Thought,  p.  178).  With  regard  to  the  first  part  of 
this  argument,  Thomson  himself  points  out  that  the  same 
objection  applies  partially  to  Figures  2  and  3.  It  is  no  doubt 
a  reason  why  as  a  matter  of  fact  Figure  4  is  seldom  used; 
but  I  cannot  see  that  it  is  a  reason  for  altogether  refusing 
to  recognise  it.  The  second  part  of  Thomson's  argument  is, 
for  a  reason  already  stated,  unsound.  The  conclusion,  for 
example,  of  Fresison  cannot  be  "the  converse  of  the  real 
conclusion,"  since  (being  an  O  proposition)  it  is  the  converse 
of  nothing  at  all. 

For  my  own  part,  I  do  not  see  how  we  can  treat  the 
syllogism  scientifically  and  completely  without  admitting 
Figure  4.  In  an  a; /;7<?r/ separation  of  figures  according  to 
the  position  of  the  terms  in  the  premisses,  it  necessarily 
appears,  and  we  find  that  valid  reasoning  may  be  made  in 


198  SYLLOGISMS.  [part  hi. 

it.     It  is  not  actually  in  frequent  use,  but  reasoning  may 
sometimes  not  unnaturally  fall  into  it ;  for  example, — 

None  of  the  apostles  were  Greeks, 
Some  Greeks  are  worthy  of  all  honour, 

therefore,   Some  worthy  of  all  honour  are  not  apostles. 

173.  The  moods  of  Figure  4  regarded  as  indirect 
moods  of  Figure  i. 

The  earliest  form  in  which  the  mnemonic  verses  ap- 
peared was  as  follows  : — ■ 

Barbara^    Ceiarent,  Darii\  FeriOj  Baralipton, 
Celantes,  Dabitis^  Fapesmo,  Frisesomorum, 
Cesare,  Camestres^  Festino,  Baroco,  Daraptt^ 
FelaptoJi,  Disamis,  Datisi^  Bocardo^  Ferison '. 

Aristotle  recognised  only  three  figures :  the  first  figure, 
which  he  considered  the  type  of  all  syllogisms  and  which 
he  called  the  perfect  figure,  the  Dictum  de  otnjii  et  nullo 
being  directly  applicable  to  it  alone ;  and  the  second  and 
third  figures,  which  he  called  imperfect  figures,  it  being 
necessary  to  reduce  them  to  Figure  i,  in  order  to  obtain  a 
test  of  their  validity. 

Before  the  fourth  figure,  however,  was  commonly  recog- 
nised as  such,  its  moods  were  recognised  in  another  form, 
namely,  as  indirect  moods  of  the  first  figure  ;  and  the  above 
mnemonics, — Baraliptony  Celantes,  Dabitis^  Fapesmo,  Frise- 
somoninij — represent  these  moods  so  regarded*. 

^  First  given  by  Petrus  Ilispanus,  afterwards  Pope  John  XXI.,  who 
died  in  1277. 

2  From  the  14th  to  the  17th  century  the  mnemonics  found  in  works 
on  Logic  usually  give  the  moods  of  Figure  4  in  this  form,  or  else  omit 
them  altogether.  Wallis  (1687)  recognises  them  in  both  forms,  giving 
two  sets  of  mnemonics. 


CHAP.  IV.] 


SYLLOGISMS. 


199 


Mansel  {Aldrich,  p.  78)  defines  an  indirect  mood  as 
"one  in  which  we  do  not  infer  the  immediate  conclusion, 
but  its  converse." 

Thus,—  All  M  is  P, 

All  .S  is  M, 

yields  the  direct  conclusion.  All  S  is  P,  in  Barbara^ 

or  the  indirect  conclusion.  Some  P  is  S,  in  Baralipton. 

Similarly  Celafites  corresponds  to  Celaretit^  and  Dabitis 
to  Darii, 

I  should  however  take  exception  to  Mansel's  definition, 
since  in  Fapcsmo  and  Frisesomoriun  we  have  indirect  moods 
to  which  there  are  no  corresponding  valid  direct  moods. 
In  these  we  do  not  infer  "the  converse  of  the  immediate 
conclusion"  since  there  is  no  immediate  conclusion.  Mansel 
deals  with  these  two  moods  very  awkwardly.  He  says, 
^^Fapesmo  and  Frisesoj?torujn  have  negative  minor  premisses, 
and  thus  offend  against  a  special  rule  of  the  first  figure; 
but  this  is  checked  by  a  counter-balancing  transgression. 
For  by  simply  converting  O,  we  alter  the  distribution  of 
the  terms,  so  as  to  avoid  an  illicit  process."  But  surely  we 
cannot  counterbalance  one  violation  of  law  by  committing 
a  second.  The  truth  of  course  is  that,  in  the  first  place, 
the  special  rules  of  Figure  i  as  ordinarily  given  do  not  apply 
to  the  indirect  moods ;  and  in  the  second  place,  the  conclu- 
sion O  is  not  obtained  by  conversion  at  all. 

The  real  distinction  between  direct  and  indirect  moods 
is  that  the  terms  which  are  major  and  minor  in  the  one 
become  respectively  minor  and  major  in  the  other. 

Taking  Ferio,  and  making  this  inversion,  we  have  no 
valid  conclusion,  therefore,  Ferio  has  no  corresponding  in- 
direct mood.     Similarly,  Fapesmo  and  Frisesomoriifn,  (the 


200 


SYLLOGISMS. 


[part  III. 


Fesapo  and  Fresison  of  Figure  4),  have  no  corresponding 
direct  moods. 

It  is  a  merely  formal  difference  whether  we  recognise 
the  five  moods  in  question  in  this  way,  or  as  constituting 
a  distinct  figure ;  but  I  think  that  the  latter  alternative  is 
far  less  likely  to  give  rise  to  confusion. 

We  have  also  indirect  moods  in  Figures  2  and  3,  but 
they  are  merely  reproductions  of  other  of  the  direct  moods, 
as  noticed  by  Professor  Fowler  (Deductive  Logic,  p.  104). 
Thus,  the  premisses, — 

No  F  is  M, 

All  5  is  AT, 

may  yield  both  the  conclusions  No  S  is  F,  and  ^0  F  is  S; 
and  if  we  call  the  former  the  direct  conclusion  the  latter 
will  be  the  indirect,  (the  former  being  Cesare,  and  the  latter 
Camestres  with  the  premisses  transposed). 

Some  F  is  M, 
No  S  is  M, 

yields  no  direct  conclusion,  but  it  has  an  indirect  conclusion 
Some  F  is  not  S.  This  however  merely  gives  Fesiino  over 
again. 

We  get  a  similar  result,  in  all  cases  in  which  this  con- 
ception is  applied  within  the  limits  of  Figures  2  and  3. 

174.  **  Rejecting  the  fourth  figure  and  the  subal- 
tern nioods,  we  may  say  with  Aristotle  ;  A  is  proved 
only  in  one  figure  and  one  mood,  E  in  two  figures 
and  three  moods,  I  in  two  figures  and  four  moods, 
O  in  three  figures  and  six  moods.  For  this  reason,  A 
is  declared  by  Aristotle  to  be  the  most  difficult  propo- 
sition to  establish,  and  the  easiest  to  overthrow;  O,  the 
reverse.*' 


SYLLOGISMS. 


201 


CHAP.  IV.] 

Discuss  the  fitness  of  these  data  to  support  the 
conclusion. 

The  above  quotation  is  from  Mansel's  Aldrich,  p.  92. 

I  do  not  think  that  the  reasoning  contained  in  it  is  sound. 

The  difficulty  of  proving  any  proposition  depends  rather  on 

the  difficulty  or  easiness  of  obtaining  premisses.     Given  the 

premisses,  it  is  not  the  more  difficult  to  prove  the  conclusion 

because  it  can  only  be  done  in  one  figure  and  one  mood, 

nor  the  easier  because  it  can  be  done  in  three  figures  and 

six  moods.    For  it  must  be  remembered  that  the  doctrine  of 

Reduction  has  shewn  us  that  the  different  moods  in  which 

O,  for  instance,  can  be  established  are  equivalent.     If  we 

were  limited  to  proving  O  in  Ferio,  we  could  do  it  just  as 

easily. 

The  above  reasoning  would  make  E  easier  to  establish 

than  A,  and  O  than  I ;  but  this  is  not  really  the  case.  If 
we  can  prove  an  E  we  can  also  prove  an  A  by  obversion ; 
and  similarly  with  O  and  I. 

For  other  reasons  it  is  of  course  obvious  that  universals 
are  more  easily  overthrown,  particulars  more  easily  esta- 
blished. A  reference  to  the  subaltern  moods  indeed  illus- 
trates this.  I  can  be  proved  in  all  cases  in  which  A  can  be 
proved,  and  some  others;  similarly  with  O  and  E. 


CHAPTER  V. 


SYLLOGISMS. 


203 


CHAP,  v.] 

those  yielded  by  the  minor.  This  gives  four  combinations, 
and  whatever  is  true  of  S  in  terms  of  F  in  a/l  of  them,  is 
the  conclusion  required. 

(i)  and  (a)  yield        (  5  ^^p  j 


THE   DIAGRAMMATIC   REPRESENTATION  OF    SYLLOGISMS. 

175.  The  application  of  the  Eulerian  diagrams 
to  Syllogistic  reasonings. 

To  illustrate  the  application  of  the  Eulerian  diagrams 
to  syllogistic  reasonings  we  may  take  a  syllogism  in  Bar- 
bara^ — 

All  ^/ is/', 
All  S  is  M, 
therefore.  All  »S  is  P. 

We  must  first  represent  the  premisses  separately  by  means 
of  the  diagrams.     They  each  yield  two  cases ;  thus, — 


AiiMisr,      (i)  [mp] 

Alls  is  M,        {a)    {    SM     \ 


(ii) 


{b) 


To  obtain  the  conclusion,  each  of  the  cases  yielded  by 
the  major  premiss  must  now  be   combined   with  each   of 


(i)  and  (b) 


(ii)  and  (a) 


(ii)  and  (h) 


We  find  that  in  all  these  cases  all  S  is  JR,  which  con- 
clusion may  therefore  be  inferred  from  the  given  premisses. 

Next,  take  a  syllogism  in  Bocardo.     The  application  of 
the  diagrams  is  now   more   complicated.      The  premisses 

are, — 

Some  J/ is  not  P, 

All  J/ is  S. 


204  SYLLOGISMS.  [part  in. 

The  major  premiss  gives  three  possible  cases, — 
(0  (")  (iii) 

0300 

and  the  minor  premiss  gives  two  possible  cases, — 


(^) 


(^) 


Taking  them  together  we  have  six  combinations,  some 
of  which  however  themselves  yield  more  than  one  case : — 


(i)  and  (a) 


(i)  and  (/;) 


\n)  and  (a) 


(ii)  and  (^) 


CHAP,  v.] 


SYLLOGISMS. 


205 


(iil)  and  (a) 


(iii)  and  (d) 


So  far  as  S  and  P  are  concerned,  (i.e.,  leaving  J/ out  of 
account),  it  will  be  found  that  these  nine  cases  are  reducible 
to  the  following  three, — 


The  conclusion  therefore  is  Some  S  is  nof  P, 

It  must  be  admitted  that  this  is  very  complex,  and  it 
would  be  a  serious  matter  if  in  the  first  instance  we  had  to 
work  through  all  the  different  moods  in  this  manner.  For 
purposes  of  illustration,  however,  this  very  complexity  has  a 
certain  advantage.  It  shews  how  many  relations  between 
three  terms  in  respect  of  extension  are  left  to  us,  even  with 
two  premisses  given. 

176.  Represent  by  means  of  the  Eulerian  dia- 
grams the  moods  Ceiarenty  Festino^  Datisi^  and  Bra- 
mantip. 


2o6 


SYLLOGISMS. 


[part  III. 

177.  What  IS  all  that  we  can  infer  by  the  aid  of 
the  following  premisses,— Some  M  is  not  Py  Some  M 
is  P,  Some  P  is  not  M,  All  M  is  5,  Some  5  is  not 

Mt 

The  first  three  premisses  limit  us  to, — 


09 


and  the  two  remaining  ones  to,- 


Combining  these,  we  have,- 


or 


that  is,  so  far  as  S  and  P  are  concerned,  the  premisses  limit 


us  to  the  following, — 


These  may  be  interpreted — 

Some  S  is  P^  Some  S  is  not  P,  and  Some  P  is  S. 

We  could  of  course  obtain  the  same  conclusions  by  the 
aid  of  the  ordinary  syllogistic  moods. 


CHAP,  v.]  SYLLOGISMS.  207 

178.    The  application  of  Mr  Venn's  diagrammatic 
scheme  to  syllogistic  reasonings. 

We  may  take  Barbara  and  Camestres  to  illustrate  the 
above. 

The  premisses  of  Barbara^ — 

All  J/  is  P, 
All  5  is  M, 

exclude  certain  compartments  as  shewn  in   the  following 
diagram, — 


Then,  so  far  as  S  and  P  are  concerned,  this  is  read  off, 
All  S  is  P. 

Similarly  for  Camestres  we  have, — 


This  scheme  is  especially  adapted  to  illustrate  the  syllo- 
gistic processes  when  all  the  propositions  involved  are  uni- 
versal. A  further  device  must  be  introduced  when  one  of 
the  premisses  is  particular.     Compare  section  95. 


208 


SYLLOGISMS. 


[part  III. 


179.  Represent  Celareiit,  Cesare,  CavicneSy  in  Mr 
Venn's  diagrammatic  scheme. 

180.  Lambert's  scheme  of  diagrammatic  notation. 

In  the  system  of  Lambert,  (slightly  altered  so  far  as 
particular  propositions  are  concerned,  compare  Venn,  Sym- 
bolic Logic  ^  p.  431),  propositions  are  represented  as  follows : — 


All  S  is  F 


No  Sis  P 


Some  S  is  P 


Some  S  is  riot  P 


P 


It  will  be  observed  that  the  extension  of  a  term  is  repre- 
sented by  a  horizontal  straight  line,  and  that  so  far  as  two 
such  lines  overlap  each  other  the  corresponding  classes  are 
coincident,  while  so  far  as  they  do  not  overlap  the  cor- 
responding classes  exclude  each  other.  The  line  is  dotted 
in  so  far  as  we  are  uncertain  with  regard  to  a  portion  of  the 
class;  />.,  a  line  representing  an  undistributed  term  is  partly 
dotted.     Thus,  in  the  case  of  All  S  is  /*, — 


the  diagram  indicates  that  all  S  is  contained  under  P,  but 
that  we  are  uncertain  as  to  whether  there  is  or  is  not  any  P 
which  is  not  S. 


CHAP,  v.] 


SYLLOGISMS. 


209 


In  the  case  of  Some  S  is  not  P^- 

P 

S 


the  diagram  indicates  that  there  is  S  which  is  not  P,  but  that 
we  are  in  ignorance  as  to  the  existence  of  any  S  that  is  P. 

181.  The  application  of  Lambert's  diagrammatic 
scheme  to  syllogistic  reasonings. 

As  applied  to  syllogisms,  the  method  indicated  in  the 
preceding  section  is  much  less  cumbrous  than  the  Eulerian 
diagrams  \     We  may  take  the  following  examples  : — 

Barbara  P 


M 


Baroco 


M 


1  Mr  Venn  {Symbolic  Logic,  p.  432)  remarks, — "As  a  whole  Lam- 
bert's scheme  seems  to  me  distinctly  inferior  to  the  scheme  of  Euler, 
and  has  in  consequence  been  very  little  employed  by  other  logicians." 
Mr  Venn's  criticism  is  chiefly  directed  against  Lambert's  representation 
of  the  particular  affirmative  proposition,  namely, — 

P 

S 

The  modification,  however,  which  I  have  here  introduced,  and  which 
is  suggested  by  Mr  Venn  himself,  meets  the  objections  raised  on  this 
ground. 


K.  L. 


14 


2IO 


Datisi 


SYLLOGISMS. 


M 


[part  III. 


Fresison 


M 


182.  Represent  the  moods  Darii^  Cesare,  Darapti, 
and  Fcsapo  in  Lambert's  scheme. 

183.  Take  the  premisses  of  an  ordinary  syllogism 
in  Barbara,  e.g.,  all  X  is  F,  all  V  is  Z ;  determine 
precisely  and  exhaustively  what  those  propositions 
affirm,  what  they  deny,  and  what  they  leave  in  doubt, 
concerning  the  relations  of  the  terms  X,  V,  Z.     [l.] 

This  question  can  be  very  well  answered  by  the  aid  of 
any  of  the  three  diagrammatic  schemes  which  we  have  just 
been  discussing.  Compare  also  Jevons,  Shidies  in  Deduc- 
tive Logic,  p.  216. 


CHAPTER  VL 


IRREGULAR    AND    COMPOUND    SYLLOGISMS. 


184.     The  Enthymeme. 

By  the  Enthymeme,  Aristotle  meant  what  has  been  called 
the  "rhetorical  syllogism"  as  opposed  to  the  apodeictic, 
demonstrative,  theoretical  syllogism.  The  following  is  from 
Mansel's  notes  to  Aldrich  (pp.  209 — 211)  :  "The  Enthy- 
meme is  defined  by  Aristotle,  (rvA.A.oyto-/xos  cf  dKojinv  tj 
(rr]fX€LOiv.  The  cikos  and  o-rjfxclov  themselves  are  Propositions ; 
the  former  stating  a  general  probability,  the  latter  a  fact, 
which  is  known  to  be  an  indication,  more  or  less  certain,  of 
the  truth  of  some  further  statement,  whether  of  a  single  fact 
or  of  a  general  belief  The  former  is  a  proposition  nearly, 
though  not  quite,  universal ;  as  'Most  men  who  envy  hate': 
the  latter  is  a  singular  proposition,  which  however  is  not 
regarded  as  a  sign,  except  relatively  to  some  other  propo- 
sition, which  it  is  supposed  may  be  inferred  from  it.  The 
€tK09,  when  employed  in  an  Enthymeme,  will  form  the  major 
premiss  of  a  Syllogism  such  as  the  following : 


therefore, 


Most  men  who  envy  hate. 

This  man  envies, 

This  man  (probably)  hates. 


14- 


212 


SYLLOGISMS. 


[part  III. 


The  reasoning  is  logically  faulty;  for,  the  major  premiss 
not  being  absolutely  universal,  the  middle  term  is  not  dis- 
tributed. 

The  (TrjfieLov  will  form  one  premiss  of  a  Syllogism  which 
may  be  in  any  of  the  three  figures,  as  in  the  following  ex- 
amples : 

Figure  i.      All  ambitious  men   are  liberal, 
Pittacus  is  ambitious, 
Therefore,  Pittacus  is  liberal. 

Figure  2.      All  ambitious  men  are  liberal, 
Pittacus  is  liberal. 
Therefore,  Pittacus  is  ambitious. 

Figicre  3.       Pittacus  is  liberal, 

Pittacus  is  ambitious, 
Therefore,  All  ambitious  men  are  liberal. 

The  syllogism  in  the  first  figure  is  alone  logically  valid. 
In  the  second,  there  is  an  undistributed  middle  term :  in 
the  third,  an  illicit  process  of  the  minor." 

On  this  subject  the  student  may  be  referred  to  the 
remainder  of  the  note  from  which  the  above  extract  is  taken, 
and  to  Hamilton,  Discussions^  pp.  152 — 156. 

An  enthymeme  is  now  usually  defined  as  a  syllogism 
incompletely  stated,  one  of  the  premisses  or  the  conclusion 
being  understood  but  not  expressed.  As  has  been  frequently 
pointed  out,  the  arguments  of  everyday  life  are  for  the  most 
part  enthymematic.  The  same  may  be  said  of  fallacious 
arguments,  which  are  seldom  completely  stated,  or  their 
want  of  cogency  would  be  more  quickly  recognised. 

An  enthymeme  is  said  to  be  of  the  first  order  when  the 
major  premiss  is  suppressed ;  of  the  second  order  when  the 
minor  premiss  is  suppressed  ;  and  of  the  third  order  when 
the  conclusion  is  suppressed. 


CHAP.  VI.] 


SYLLOGISMS. 


213 


Thus,  "Balbus  is  avaricious,  and  therefore,  he  is  un- 
happy," is  an  enthymeme  of  the  first  order ;  "  All  avaricious 
persons  are  unhappy,  and  therefore,  Balbus  is  unhappy"  is 
an  enthymeme  of  the  second  order;  "All  avaricious  persons 
are  unhappy,  and  Balbus  is  avaricious  "  is  an  enthymeme  of 
the  third  order. 

185.     The  Polysyllogism  ;  and  the  Epicheirema. 

A  chain  of  syllogisms,  that  is,  a  series  of  syllogisms  so 
linked  together  that  the  conclusion  of  one  becomes  a  pre- 
miss of  another,  is  called  a  polysyllogism.  In  a  polysyllogism, 
any  individual  syllogism  the  conclusion  of  which  becomes 
the  premiss  of  a  succeeding  one  is  called  a  prosyllogism ; 
any  individual  syllogism  one  of  the  premisses  of  which  is 
the  conclusion  of  a  preceding  syllogism  is  called  an  epi- 
syllogisnu     Thus, — 

All  C  is  Z>, ) 

All  B\^c\    prosyllogism, 

therefore.  All  B  is  Z>,  ? 

but.  All  A  is  ^,  I     episyllogism. 
therefore.  All  A  is  Z>. ) 

The  same  syllogism  may  of  course  be  both  an  episyllo- 
gism and  a  prosyllogism,  as  would  be  the  case  with  the 
above  episyllogism  if  the  chain  were  continued  further. 

An  epicheire7na  is  a  polysyllogism  with  one  or  more 
prosyllogisms  briefly  indicated  only.  That  is,  one  or  more 
of  the  syllogisms  of  which  the  polysyllogism  is  composed 
is  enthymematic.  Whately  {Logic,  p.  117)  calls  it  accord- 
ingly an  enthymetnatic  sentence.    The  following  is  an  example, 

B  is  Z>,  because  it  is  C, 
^  is  ^, 
therefore,  A  is  D. 


214 


SYLLOGISMS. 


[part  III. 


186.  The  Sorites. 

A  Sorites  is  a  polysyllogism  in  which  all  the  conclusions 
are  omitted  except  the  final  one;  for  example, 

A  isB, 
Bis  C, 
CisZ>, 
jD  is  ^, 
therefore,  A  is  E. 

187.  The  ordinary  Sorites,  and  the  Goclenian 
Sorites. 

In  the  ordinary  Sorites,  the  premiss  which  contains  the 
subject  of  the  conclusion  is  stated  first;  in  the  Goclenian 
Sorites  it  is  stated  last.    Thus, — 

Ordinary  Sorites^ —      A  is  B, 

Bis  C, 
CisZ>, 
DisE, 

therefore,  A  is  E. 

Goclenian  Sorites, —      D  is  E, 

CisD, 

Bis  Q 

AisB, 

therefore,  A  is  E. 

If,  in  the  case  of  the  ordinary  sorites,  the  argument  were 
drawn  out  in  full,  the  suppressed  conclusions  would  appear 
as  minor  premisses  in  successive  syllogisms.  Thus,  the 
ordinary  sorites  given  above  may  be  analysed  into  the  three 
following  syllogisms, — 

(i)  B  is  C, 

^is^, 
therefore,  ^  is  C; 


CHAP.  VI.] 
(2) 


SYLLOGISMS. 


215 


Cis  A 

^  is  C, 
therefore,  A  is  jD; 

(3)  ^  is  E, 

A  is  JD, 
therefore,  A  is  E. 

Here  the  suppressed  conclusion  of  (i)  is  seen  to  be  the 
minor  premiss  of  (2),  that  of  (2)  the  minor  premiss  of  (3); 
and  so  it  would  go  on  if  the  number  of  propositions  con- 
stituting the  Sorites  were  increased. 

In  the  Goclenian  Sorites,  the  premisses  are  the  same, 
but  their  order  is  reversed,  and  the  result  of  this  is  that 
the  suppressed  conclusions  become  major  premisses  in 
successive  syllogisms. 

Thus  the  Sorites,—      B  is  E, 

CisD, 

Bis  Cy 

A  is  By 

therefore,  A  is  E, 

may  be  analysed  into  the  following  three  syllogisms, — 

(i)  ^  is  E, 

CisD, 

therefore,  C  is  ^ ; 

(2)  C  is  E, 

Bis  C, 
therefore.  Bis  E; 

(3)  ^    is     Ey 

A  is  By 
therefore,  A  is  E. 

Here  the  conclusion  of  (i)  becomes  the  major  premiss 

of  (2);  and  so  on. 


2l6 


SYLLOGISMS. 


[part  III. 


The  ordinary  Sorites'  is  that  which  is  most  usually 
discussed;  but  it  may  be  noted  that  the  order  of  premisses 
in  the  Goclenian  form  is  that  which  really  corresponds  to 
the  customary  order  of  premisses  in  a  simple  syllogism. 

188.     The  special  rules  of  the  ordinary  Sorites. 

The  special  rules  of  the  ordinary  sorites  are, — 

(i)  Only  one  premiss  can  be  negative;  and  if  one  is 
negative,  it  must  be  the  last. 

(2)  Only  one  premiss  can  be  particular;  and  if  one  is 
particular,  it  must  be  the  first. 

Any  ordinary  sorites  may  be  represented  in  skeleton 
form,  the  quantity  and  quality  of  the  premisses  being  left 
undetermined,  as  follows  : — 

1  What  I  have  called  the  ordinary  Sorites  is  frequently  spoken  of  as 
the  Aristotelian  Sorites ;  for  example,  by  Archbishop  Thomson  {Laws 
of  Thought,  p.  201),  and  Spalding  {Logic,  p.  302).     Hamilton  however 
remarks, — "The  name  Sorites  does  not  occur  in  any  logical  treatise 
of  Aristotle ;    nor,  as  far  as  I  have  been  able  to  discover,  is  there, 
except  in  one  vague  and  cursory  allusion,  any  reference  to  what  the 
name  is  now  employed  to  express"  {Lectures  on  Logic,  i.  p.  375).     The 
term  Sorites  (from  o-wpis,  a  heap)  as  used  by  ancient  writers  was  em- 
ployed to  designate  a  particular  sophism,  based  on  the  difficulty  which 
is  sometimes  found  in  assigning  an  exact  limit  to  a  notion.     "  It  was 
asked,— was  a  man  bald  who  had  so  many  thousand  hairs  j  you  answer, 
No  :  the  antagonist  goes  on  diminishing  and  diminishing  the  number, 
till  either  you  admit  that  he  who  was  not  bald  with  a  certain  number  of 
hairs,  becomes  bald  when  that  complement  is  diminished  by  a  single 
hair ;  or  you  go  on  denying  him  to  be  bald,  until  his  head  be  hypotheti- 
cally  denuded." 

The  distinct  exposition  of  the  kind  of  reasoning  which  is  now  known 
as  the  Sorites  is  attributed  to  the  Stoics ;  but  it  was  not  called  by  this 
name  till  the  fifteenth  century  (Hamilton,  Logic,  i.  p.  377).  The  form 
of  Sorites  called  the  Goclenian  was  "first  given  ])y  Goclenius  in  his 
Isagoge  in  Organum  Aristotelis,  1598"  (Hansel's  Aldrich,  p.  96). 


CHAP.  VI.] 


SYLLOGISMS. 

S  M^ 


217 


M  , 

M 

rt-2 

n~ 

^„-. 

M, 

M„ 

P 

S  P 

(i)  There  cannot  be  more  than  one  negative  premiss, 
for  if  there  were,  (since  a  negative  premiss  in  any  syllogism 
necessitates  a  negative  conclusion),  we  should  in  analysmg 
the  sorites  somewhere  come  upon  a  syllogism  containing 
two  negative  premisses. 

Again,  if  one  premiss  is  negative,  the  final  conclusion 
must  be  negative.  Therefore,  P  must  be  distributed  in  this 
conclusion.  Therefore,  it  must  be  distributed  in  its  premiss, 
/>.,  the  last  premiss,  which  must  therefore  be  negative. 

If  any  premiss  then  is  negative,  this  is  the  one. 

(2)  Since  it  has  been  shewn  that  all  the  premisses, 
except  the  last,  must  be  affirmative,  it  is  clear  that  if  any, 
except  the  first,  were  particular,  we  should  somewhere 
commit  the  fallacy  of  undistributed  middle. 

189.  Find  and  prove  the  special  rules  of  the 
Goclenian  Sorites. 

190.  The  possibility  of  a  Sorites  in  a  Figure 
other  than  the  First. 

It  will  have  been  noticed  that  in  analysing  both  the 
(so-called)  Aristotelian  and  the  Goclenian  Sorites  all  the 
resultant  syllogisms  are  in  Figure  i.  Such  sorites  therefore 
may  themselves  be  said  to  be  in  Figure  i.  The  question 
arises  whether  a  sorites  is  possible  in  any  other  figure. 


2l8 


SYLLOGISMS. 


[part  III. 


Sir  William  Hamilton  {Lectures  on  Logic,  vol.  2,  p.  403) 
remarks  that  "  all  logicians  have  overlooked  the  Sorites  of 
Second  and  Third  Figures."     Reading  on,  however,  we  find 
that  by  a  Sorites  in  the  Second  Figure  he  means  such  a 
reasoning  as  the  following ;— No  B  is  A,  No  C  is  A,  No  L> 
is  A,  No  E  is  A,  All  F  is  A,  therefore.  No  B,  or  C,  or  L> 
or  ^,  is  i^;  and  by  a  Sorites  in  the  Third  Figure  such  as  the 
following  '.—A  is  B,  A  is  Q  A  is  D,  A  \s  E,  A  is  E,  there- 
fore, Some  ^,  and  C,  and  Z>,  and  E,  are  /^     (He  does  not 
himself  give  these  examples;  but  that  this  is  what  he  means 
may  be  deduced  from  his  not  very  lucid  statement, — "  In 
Second  and  Third  Figures,  there  being  no  subordination  of 
terms,  the  only  Sorites  competent  is  that  by  repetition  of  the 
same  middle.     In  First  Figure,  there  is  a  new  middle  term 
for  every  new  progress  of  the  Sorites  ;  in  Second  and  Third, 
only  one  middle  term  for  any  number  of  extremes.     In 
First  Figure,  a  syllogism  only  between  every  second  term  of 
the  Sorites,  the  intermediate  term  constituting  the  middle 
term.     In  the  others,  every  two  propositions  of  the  common 
middle  term  form  a  syllogism.") 

But  it  is  clear  that  in  the  accepted  sense  of  the  term 
these  are  not  sorites  at  all.  In  neither  of  them  have  we  any 
chain  argument,  but  our  conclusion  is  a  mere  summation  of 
the  conclusions  of  a  number  of  syllogisms  having  a  common 
premiss. 

Hamilton's  own  definition  of  sorites,  involved  as  it  is, 
might  have  saved  him  from  this  error.  He  gives  for  his 
definition, — "  When,  on  the  common  principle  of  all  reason- 
ing,— that  the  part  of  a  part  is  a  part  of  the  whole, — we  do 
not  stop  at  the  second  gradation,  or  at  the  part  of  the  high- 
est part,  and  conclude  that  part  of  the  whole,  but  proceed 
to  some  indefinitely  remoter  part,  as  D,  E,  E,  G,  H,  &c., 
which,  on  the  general  principle,  we  connect  in  the  conclusion 


CHAP.  VI.] 


SYLLOGISMS. 


219 


with  its  remotest  whole, — this  complex  reasoning  is  called 
a  Chain-Syllogisfn  or  Sorites''  {Lectures  07i  Logic^  vol  i. 
p.  366). 

In  the  above  criticism  I  have  followed  J.  S.  MilP.  His 
own  treatment  of  the  question,  however,  seems  open  to 
refutation  by  the  simple  method  of  constructing  examples. 
He  considers  that  the  first  or  last  syllogism  of  a  sorites 
may  be  in  Figure  2  or  3,  {e.g.,  in  Figure  2  we  might  have 
A'\sB,B  is  C,  C'lsD,  D  is  E,  No  F  is  E,  therefore  A  is 
not  F\  but  that  it  is  impossible  that  all  the  steps  should  be 
in  either  of  these  figures.  "  Every  one  who  understands  the 
laws  of  the  second  and  third  figures  (or  even  the  general  laws 
of  the  syllogism)  can  see  that  no  more  than  one  step  in 
either  of  them  is  admissible  in  a  sorites,  and  that  it  must 
either  be  the  first  or  the  last." 

But  take  the  following  (the  suppressed  conclusions  being 
inserted  in  square  brackets) : — 

All  A  is  B, 
No  Cis  B, 
[therefore.  No  A  is  C], 

All  D  is  C, 
[therefore,  No  A  is  D\ 

All  E  is  D, 
[therefore,  No  A  is  E\ 

AllEisE, 
therefore.  No  A  is  E^, 

^  In  connection  with  it,  Mill  very  justly  remarks, — "  If  Sir  W. 
Hamilton  had  found  in  any  other  writer  such  a  misuse  of  logical 
language  as  he  is  here  guilty  of,  he  would  have  roundly  accused  him  of 
total  ignorance  of  logical  writers"  {Exatnination  of  Hamilton,  p.  515). 

2  This  Sorites  is  analogous  to  the  so-called  Aristotelian  Sorites, 
the  subject  of  the  conclusion  appearing  in  the  premiss  stated  first.  It  is 
to  be  observed  that  the  rules  given  in  section  188  will  not  apply  except 


220 


SYLLOGISMS. 


[part  III. 


All  the  syllogisms  involved  here  are  in  Figure  2,  and  the 
sorites  itself  may  I  think  fairly  be  said  to  be  in  Figure  2.  As 
in  the  ordinary  sorites,  the  conclusion  of  each  syllogism  is 
the  minor  of  the  next. 

The  following  again  may  be  called  a  sorites  in  Figure  3 : — 

All  B  is  A, 
All  B  is  C, 
[therefore,  Some  C  is  A\ 

All  C  is  D, 
[therefore,  Some  D  is  A\ 

All  D  is  E, 
therefore,  Some  E  is  A^ 

therefore,  So7ne  A  is  E\ 

Here  the  conclusion  of  each  syllogism  is  the  major  of  the 
next  ^ 

191.  Take  any  Enthymeme  (in  the  modern  sense) 
and  supply  premisses  so  as  to  expand  it  into  {a)  a 
syllogism,  {b)  an  epicheirema,  {c)  a  sorites;  and  name 
the  mood,  order  or  variety  of  each  product.  [c] 

192.  Is  there  any  case  in  which  a  conclusion  can 
be  obtained  from  two  premisses,  although  the  middle 
term  is  distributed  in  neither  of  them  } 

The  ordinary  syllogistic  rule  relating  to  the  distribution 
of  the  middle  term  does  not  contemplate  the  recognition  of 

when  the  Sorites  is  in  Figure  i.  For  Sorites  in  Figures  2  and  3,  how- 
ever, other  rules  might  be  framed  corresponding  to  the  special  rules  of 
Figures  2  and  3  in  the  case  of  the  simple  Syllogism. 

^  The  preceding  note  applies  to  this  Sorites  also. 

^  I  should  admit  that  such  Sorites  as  the  above  are  not  likely  to  be 
found  in  common  use. 


CHAP.  VI.] 


SYLLOGISMS. 


221 


any  signs  of  quantity  other  than  all  and  some.  The  admis- 
sion of  the  sign  inosi  yields  the  valid  reasoning, — 

Most  M  is  P, 

Most  M  is  Sy 

therefore.  Some  S  is  B. 

We  understand  most  in  the  sense  of  more  than  half^  and  it 
clearly  follows  from  the  above  premisses  that  there  must  be 
some  M  which  is  both  »S  and  B.  We  cannot  however  say 
that  in  either  premiss  the  term  M  is  distributed.  To  meet 
this  case,  then,  the  rule  with  regard  to  the  distribution  of  the 
middle  term  must  be  amended,  if  other  signs  of  quantity 
besides  all  and  some  are  recognised. 

Sir  W.  Hamilton  (Logic^  vol.  2,  p.  362)  gives, — "The 
quantifications  of  the  middle  term,  whether  as  subject  or 
predicate,  taken  together,  must  exceed  the  quantity  of  that 
term  taken  in  its  whole  extent";  in  other  words,  we  require 
the  ultra-total  distribution  of  the  middle  term,  in  the  two 
premisses  taken  together.  Hamilton  then  continues  some- 
what too  dogmatically, — "  The  rule  of  the  logicians,  that  the 
middle  term  should  be  once  at  least  distributed,  is  untrue. 
For  it  is  sufficient  if,  in  both  the  premisses  together,  its 
quantification  be  more  than  its  quantity  as  a  whole,  (ultra- 
total).  Therefore,  a  major  party  (a  viore  or  7nost\  in  one 
premiss,  and  a  half  in  the  other,  are   sufficient  to  make 

it  effective." 

De  Morgan  {Bormal  Logic,  p.  127)  writes  as  follows, — 
"  It  is  said  that  in  every  syllogism  the  middle  term  must  be 
universal  in  one  of  the  premisses,  in  order  that  we  may 
be  sure  that  the  affirmation  or  denial  in  the  other  premiss 
may  be  made  of  some  or  all  of  the  things  about  which 
affirmation  or  denial  has  been  made  in  the  first.  This  law, 
as  we  shall  see,  is  only  a  particular  case  of  the  truth:  it 


222 


SYLLOGISMS. 


[part  III. 


is  enough  that  the  two  premisses  together  affirm  or  deny  of 
more  than  all  the  instances  of  the  middle  term.  If  there  be 
a  hundred  boxes,  into  which  a  hundred  and  one  articles 
of  two  different  kinds  are  to  be  put,  not  more  than  one 
of  each  kind  into  any  one  box,  some  one  box,  if  not  more, 
will  have  two  articles,  one  of  each  kind,  put  into  it.  The 
common  doctrine  has  it,  that  an  article  of  one  particular 
kind  must  be  put  into  every  box,  and  then  some  one  or 
more  of  another  kind  into  one  or  more  of  the  boxes,  before 
it  may  be  affirmed  that  one  or  more  of  different  kinds  are 
found  together."  De  Morgan  himself  works  the  question 
out  in  detail  in  his  treatment  of  the  numerically  definite 
syllogism^  {Formal  Logic,  pp.  141 — 170). 

193.  The  Argument  a  fortiori  and  other  de- 
ductive inferences  that  are  not  reducible  to  the 
ordinary  syllogistic  form. 

We  may  take  as  an  example  of  the  argument  a  fortiori: 

B  is  greater  than  (7, 

A  is  greater  than  B, 

therefore,  A  is  greater  than  C. 

As  this  stands,  it  is  clearly  not  in  the  ordinary  syllogistic 
form  since  it  contains  four  terms ;  some  logicians  however 
profess  to  reduce  it  to  the  ordinary  syllogistic  form  as 
follows : 

B  is  greater  than  C, 
therefore,  a  greater  than  B  is  greater  than  C, 
but,  ^  is  a  greater  than  B, 

therefore,  A  is  greater  than  C 

With  De  Morgan,  we  may  treat  this  as  a  mere  evasion, 
or  as  a  petitio  priiicipii.  The  principle  of  the  argument 
a  fortiori  is  really  assumed  in  passing  from  "  B  is  greater 
than  C"  to  "a  greater  than  B  is  greater  than  C" 


CHAP.  VI.] 


SYLLOGISMS. 


223 


The  following  attempted  resolution^  must  I  think  be 
disposed  of  similarly : 

Whatever  is  greater  than  a  greater  than  C  is  greater 
than  C, 

A  is  greater  than  a  greater  than  (7, 
therefore,  A  is  greater  than  C. 

At  any  rate,  it  is  clear  that  this  cannot  be  the  whole  of 
the  reasoning,  since  B  no  longer  appears  in  the  premisses 
at  all. 

Mansel  {Aldrichy  pp.  199,  200)  treats  the  argument 
a  fortiori  diS  a  material  coiiseqiience,  and  by  this  he  means,  "  one 
in  which  the  conclusion  follows  from  the  premisses  solely  by 
the  force  of  the  terms,"  i.e.,  "from  some  understood  pro- 
position or  propositions,  connecting  the  terms,  by  the 
addition  of  which  the  mind  is  enabled  to  reduce  the  conse- 
quence to  logical  form."  He  would  reduce  the  argument 
a  fortiori  in  one  of  the  ways  already  referred  to.  This 
however  begs  the  question  that  the  syllogistic  is  the  only 
logical  form.  As  a  matter  of  fact  the  cogency  of  the  argu- 
ment a  fortiori  is  just  as  intuitively  evident  as  that  of  a 
syllogism  in  Barbara  itself.  Why  should  no  relation  be 
regarded  as  formal  unless  it  can  be  expressed  by  the  word 
isi  Touching  on  this  case,  De  Morgan  remarks  that  the 
formal  logician  has  a  right  to  confine  himself  to  any  part  of 
his  subject  that  he  pleases  j  *'but  he  has  no  right  except 
the  right  of  fallacy  to  call  that  part  the  whole  "  (Syllabus, 
p.  42). 

"-A  equals  B\  B  equals  C\  therefore,  A  equals  (7"  is 
another  case  to  which  the  same  remarks  apply. 

"  This  is  not  an  instance  of  common  syllogism  :  the 
premisses  are  ^A  is  an  equal  oi  B  \  B  is  an  equal  of  C   So 


1  Cf.  Mansel's  Aldrich,  p,  200. 


224 


SYLLOGISMS. 


[PART  III. 


far  as  common  syllogism  is  concerned,  that  *an  equal  o{  B' 
is  as  good  for  the  argument  as  *^'  is  a  material  accident  of 
the  meaning  of  *  equal.'  The  logicians  accordingly,  to  re- 
duce this  to  a  common  syllogism,  state  the  effect  of  com- 
position of  relation  in  a  major  premiss,  and  declare  that 
the  case  before  them  is  an  example  of  that  composition  in 
a  minor  premiss.  As  in,  A  is  aft  equal  of  an  equal  (of  C); 
Every  equal  of  an  equal  is  an  equal;  therefore,  A  is  an  equal  oi 
C.  This  I  treat  as  a  mere  evasion.  Among  various  suffi- 
cient answers  this  one  is  enough :  men  do  not  think  as 
above.  When  A  =  B,  B  =  C,  is  made  to  give  A=  Cy  the 
word  equals  is  a  copiila  in  thought,  and  not  a  notion  attached 
to  a  predicate.  There  are  processes  which  are  not  those  of 
common  syllogism  in  the  logician's  major  premiss  above  : 
but  waiving  this,  logic  is  an  analysis  of  the  form  of  thought, 
possible  and  actual,  and  the  logician  has  no  right  to  declare 
that  other  than  the  actual  is  actual."  (De  Morgan,  Syllabus ^ 

PP-  31,  2-) 

There   are  an   indefinite   number   of  other  arguments 

which  for  similar  reasons  cannot  be  reduced  to  syllogistic 
form.  For  example, — X  is  a  contemporary  of  K,  and 
V  of  Z;  therefore  Jf  is  a  contemporary  of  Z.  A  is  the 
brother  of  B,  B  is  the  brother  of  C ;  therefore,  A  is  the 
brother  of  C. 

We  must  then  reject  the  claims  that  have  been  put  for- 
ward on  behalf  of  the  syllogism  to  be  the  exclusive  form  of 
all  deductive  reasoning. 

As  an  example  of  such  claims  being  made,  Whately  may 
be  quoted.  Syllogism,  he  says,  is  "the  form  to  which  all 
correct  reasoning  may  be  ultimately  reduced  "  (Zogic,  p.  1 2). 
Again,  he  remarks,  "An  argument  thus  stated  regularly  and 
at  full  length,  is  called  a  Syllogism ;  which  therefore  is 
evidently  not  a  peculiar  kind  of  argume7it^  but  only  a  peculiar 


CHAP.  VI.] 


SYLLOGISMS. 


225 


form  of  expression,  in  which  every  argument  maybe  stated" 
{Logic,  p.  26)  ^ 

Spalding  seems  to  have  the  same  thing  in  view  when  he 
says, — "An  inference,  whose  antecedent  is  constituted  by 
more  propositions  than  one,  is  a  Mediate  Inference.  The 
simplest  case,  that  in  which  the  antecedent  propositions  are 
two,  is  the  Syllogism.  The  syllogism  is  the  norm  of  all 
inferences  whose  antecedent  is  more  complex ;  and  all  such 
inferences  may,  by  those  who  think  it  worth  while,  be  resolved 
into  a  series  of  syllogisms"  {Logic,  p.  158). 

J.  S.  Mill  endorses  these  claims.  He  remarks, — "All 
valid  ratiocination ;  all  reasoning  by  which  from  general 
propositions  previously  admitted,  other  propositions  equally 
or  less  general  are  inferred ;  may  be  exhibited  in  some  of 
the  above  forms,"  i.e.,  the  syllogistic  moods,  {Logic,  i. 
p.  191). 

What  is  required  to  fill  the  logical  gap  which  is  created 
by  the  admission  that  the  syllogism  is  not  the  norm  of  all 
valid  formal  inference  has  been  called  the  Logic  of  Rela- 
tives. The  function  of  the  Logic  of  Relatives  is  to  "take 
account  of  relations  generally,  instead  of  those  merely  which 
are  indicated  by  the  ordinary  logical  copula  is^\  (Venn, 
Symbolic  Logic,  p.  400).  The  line  which  this  new  Logic  is 
likely  to  take,  if  it  is  ever  fully  worked  out,  is  indicated  by 
the  following  passage  from  De  Morgan  {Syllabus,  pp.  30, 

31):— 

"A  convertible  copula  is  one  in  which  the  copular  rela- 
tion exists  between  two  names  both  ways :  thus  *is  fastened 
to,'  *is  joined  by  a  road  with,'  *is  equal  to,'  *is  in  habit  of 
conversation  with,'  &c.  are  convertible  copulae.  If  *^is  equal 
to  F'  then  *F  is  equal  to  X^  &c.  A  transitive  Q,Q^^Az.  is  one 
in  which  the  copular  relation  joins  X  with  Z  whenever  it 


^  Cf.  also  Whately,  Logic,  pp.  24,  5,  and  p.  34. 


K.  L. 


15 


226 


SYLLOGISMS. 


[part  III. 


joins  X  with  Y  and  Y  with  Z.  Thus  *is  fastened  to'  is 
usually  understood  as  a  transitive  copula  :  *^  is  fastened  to 
Y'  and  'Y  is  fastened  to  Z'  give  'X  is  fastened  to  Z'  All 
the  copulae  used  in  this  syllabus  are  transitive.  The  intran- 
sitive copula  cannot  be  treated  without  more  extensive 
consideration  of  the  combination  of  relations  than  I  have 
now  opportunity  to  give :  a  second  part  of  this  syllabus  or 
an  augmented  edition,  may  contain  something  on  this  sub- 
ject." The  Student  may  further  be  referred  to  Venn, 
Symbolic  Logic,  pp.  399 — 404. 


CHAPTER   VIL 


HYPOTHETICAL   SYLLOGISMS. 


194.  The  Hypothetical  Syllogism  and  the  Hypo- 
thetico-Categorical  Syllogism. 

The  form  of  reasoning  in  which  a  hypothetical  conclusion 
is  inferred  from  two  hypothetical  premisses  is  apparently  over- 
looked by  some  logicians  ;  at  any  rate,  it  frequently  receives 
no  distinct  recognition,  the  term  "  hypothetical  syllogism  " 
being  limited  to  the  case  in  which  one  premiss  only  is 
hypothetical. 

I  should  however  prefer  the  following  definitions : — 

A  Hypothetical  Syllogism  is  a  mediate  reasoning  consist- 
ing of  three  propositions  in  which  both  the  premisses  and 
the  conclusion  are  hypothetical  in  form ; 

e.  g., — If  C  is  D,  E  is  F, 

If  A  is  B,  C  is  n, 

therefore,  If  A  is  B,  E  is  F, 

A  Hypothetico-  Categorical  Syllogism  is  a  mediate  reason- 
ing consisting  of  three  propositions  in  which  one  of  the 
premisses  is  hypothetical  in  form,  while  the  other  premiss 
and  the  conclusion  are  categorical ; 

e.  g., — If  A  is  B,  C  is  D, 
A  is  B, 
therefore,  C  is  D. 

15—2 


228 


SYLLOGISMS. 


[part  III. 


This  nomenclature  is  adopted  by  Spalding  and  Ueber- 
weg,  but,  as  I  have  already  hinted,  it  is  not  the  most  usual. 
Some  logicians,  {e.g.f  Fowler),  call  either  of  the  above  forms 
of  reasoning  hypothetical  syllogisms  without  distinction. 
Others,  {e.  g.,  Jevons),  define  the  hypothetical  syllogism  so 
as  to  include  the  latter  form  alone,  the  fonner  apparently 
not  being  regarded  by  them  as  a  distinct  form  of  reasoning 
at  all.  This  view  may  be  to  some  extent  justified  by  the 
very  close  analogy  that  exists  between  the  syllogism  with 
two  hypothetical  premisses  and  the  categorical  syllogism ; 
but  the  difference  in  form  is  worth  at  least  a  brief  discussion. 

The  student  should  however  bear  in  mind  that  by  the 
"hypothetical  syllogism"  in  most  English  works  on  Logic  is 
meant  what  has  been  defined  above  as  the  hypothetico- 
categorical  syllogism. 

195.  Distinction  of  Figure  and  Mood  in  the  case 
of  Hypothetical  Syllogisms. 

In  the  Hypothetical  Syllogism,  (as  defined  in  the  pre- 
ceding section),  the  antecedent  of  the  conclusion  is  equiva- 
lent to  the  minor  term  of  the  categorical  syllogism,  the 
consequent  of  the  conclusion  to  the  major  term,  and  the 
element  which  does  not  appear  in  the  conclusion  at  all  to 
the  middle  term.  Distinctions  of  mood  and  figure  may  be 
recognised  in  precisely  the  same  way  as  in  the  case  of  the 
categorical  syllogism.     For  example, — 

Barbara, — If  C  is  Z>,  E  is  F, 

If  A  is  B,  Cis  n, 

therefore.  If  A  is  B,  E  is  F. 

Festino, —  If  E  is  F,  C  is  not  D. 
In  some  cases  in  which  A  is  B,  C  is  Z>, 
therefore,  In  some  cases  in  which  A  is  B,  E  is  not  F. 


CHAP.  VII.] 


SYLLOGISMS. 


229 


Darapti—If  C  is  Z>,  E  is  F, 
If  C  is  D,  A  is  B, 
therefore.  In  some  cases  in  which  A  is  B,  E  is  F, 

Camenes, — If  E  is  F,  C  is  D, 

If  C  is  Dy  A  is  not  B, 
therefore.  If  A  is  B,  E  is  not  F. 

In  working  with  hypotheticals  it  must  always  be  remem- 
bered that  the  quality  of  the  proposition  is  determined  by 
the  quality  of  the  consequent. 

196.  The  Reduction  of  Hypothetical  Syllo- 
gisms. 

Hypothetical  Syllogisms  in  Figures  2,  3,  4  may  be  re- 
duced to  Figure  i  just  as  in  the  case  of  Categorical  Syllo- 
gisms. Thus,  the  syllogism  in  Camenes  given  in  the  preceding 
example  is  reduced  as  follows  to  Camestres, — 

If  C  is  Z>,  A  is  not  B, 

If  E  is  F,  C  is  Z>, 
therefore,  If  E  is  F,  A  is  not  B, 
therefore,  If  A  is  B,  E  is  not  F. 

According  to  rule,  the  premisses  have  here  been  trans- 
posed, and  the  conclusion  of  the  new  syllogism  is  converted 
in  order  to  obtain  the  original  conclusion. 

197.  Construct  Hypothetical  Syllogisms  in  Cesare, 
Bocardo,  Fesapo,  and  reduce  them  to  Figure  i. 

198.  Name  the  mood  and  figure  of  the  following : 

(i)     If  C  is  D,  E  is  not  F, 

In  some  cases  in  which  A  is  By  C  is  D^ 

therefore,  /;/  some  cases  in  which  A  is  B,  E  is  not  F. 


230 


SYLLOGISMS. 


[part  III. 


(2)     IfEisF.CisD, 
If  C  is  D,  A  is  B, 
therefore,  In  some  cases  in  which  A  is  B,  E  is  F, 

Shew  that  one  of  these  forms  may  be  indirectly- 
reduced  to  the  other,  but  not  vice  versa.  Why  is 
this  ? 

199.  Name  the  mood  and  figure  of  the  follov/ing, 
and  shew  that  either  one  may  be  reduced  to  the  other 
form : — 

(i)    If  Bis  not  FX  is  D, 
If  A  is  B,  Cisnot  D, 
therefore,  If  A  is  B,  E  is  F. 

(2)    IfCisD.EisnotF, 
If  A  is  not  By  C  is  D, 
therefore.  If  A  is  not  B,  E  is  7wt  F. 

200.  The  Moods  of  the  Hypothetico-categorical 
Syllogism. 

It  is  usual  to  distinguish  two  moods  of  the  hypothetico- 
categorical  syllogism : 

(i)  The  modus ponens,  (also  called  the  constructive  hypo- 
thetical syllogism),  in  which  the  categorical  premiss  affirms 
the  antecedent  of  the  hypothetical  premiss,  thereby  justifying 
as  a  conclusion  the  affirmation  of  its  consequent.  For  ex- 
ample,— 

If  A  is  B,  A  is  C 
A  is  B, 
therefore,   A  is  C. 

(2)     The  modus  tollens,  (also  called  the  destructive  hypo- 
thetical syllogism),  in  which  the  categorical  premiss  denies 


SYLLOGISMS. 


231 


CHAP.  VII.] 

the  consequent  of  the  hypothetical  premiss,  thereby  justify- 
ing as  a  conclusion  the  denial  of  its  antecedent.  For  ex- 
ample,— 

If  A  is  By  A  is  (7, 
A  is  not  C, 
therefore,   A  is  not  B, 
These  may  be  considered  to  correspond  respectively  to 
Figures  i  and  2  of  the  categorical  syllogism. 

Thus,  the  example  of  modus ponens  given  above  may  be 
written, — 

All  cases  of  A  being  B  are  cases  of  A  being  C, 
This  case  of  A  is  a  case  of  A  being  B, 
therefore,      This  case  of  A  is  a  case  of  A  being  C; 
and  we  then  have  a  syllogism  in  Barbara, 

The  following  corresponds  to  Celarenty — 

If  A  is  By  A  is  not  C, 
A  is  B, 
therefore,  A  is  not  C, 
The  example  of  modus  tolle?is  given  above  corresponds 
to  Camestres.     The  following  corresponds  to  CesarCy— 

If  A  is  By  A  is  not  C, 
A  is  C, 
therefore,   A  is  not  B. 

201.    Reduction  of  the  7nodtis  tollens  to  the  modus 
ponens. 

Any  case  of  modus  tollens  may  be  reduced  to  modus 
ponens  and  vice  versa. 

Thus,  If  A  is  B,  A  is  C, 

A  is  not  C, 
therefore,   A  is  not  By 


232 


SYLLOGISMS. 


[part  III. 


becomes  by  contraposition  of  the  hypothetical  premiss, 

If  A  is  not  C,  A  is  not  B, 
A  is  ?iot  C 
therefore,   A  is  not  B ; 
and  this  is  modus  ponens. 

It  may  be  worth  noticing  here  that  a  categorical  syl- 
logism in  Camestres  may  similarly  be  reduced  to  Celarent 
without  transposing  the  premisses: — 

All  P  is  M, 

No  S  is  M, 

therefore,   No  S  is  P. 

No  not-M  is  P, 
All  S  is  not-M, 
therefore,   No  S  is  P, 

202.  Shew  how  the  modtis ponens  may  be  reduced 
to  the  modus  tollens, 

203.  Mention  two  fallacious  modes  of  arguing 
from  a  hypothetical  major  premiss.  To  what  falla- 
cies in  categorical  syllogisms  do  they  respectively 
correspond  ?  [c] 

There  are  two  principal  fallacies  to  which  we  are  liable 
in  arguing  from  a  hypothetical  major  premiss: — 

(i)  It  is  a  fallacy  if  we  regard  the  affirmation  of  the 
consequent  as  justifying  the  affirmation  of  the  antecedent. 
For  example, 

If  A  is  B,  A  is  C, 
A  is  C, 
therefore,    A  is  B  \ 

^  This  would  of  course  be  no  longer  a  fallacy  if  A  is  B  were  given 
as  the  sole  condition  of  A  is  C. 


CHAP.  VII.] 


SYLLOGISMS. 


233 


(2)  It  is  a  fallacy  if  we  regard  the  denial  of  the  antece- 
dent as  justifying  the  denial  of  the  consequent.  For  ex- 
ample, 

If  A  is  B,A  is  C, 

A  is  not  B, 
therefore,  A  is  not  C\ 

It  will  easily  be  seen  that  these  correspond  respectively 
to  undistributed  middle  and  illicit  major  in  the  case  of  cate- 
gorical syllogisms. 

204.  The  claims  of  the  Hypothetico-categorical 
Syllogism  to  be  regarded  as  Mediate  Inference. 

Taking  the  syllogism, — 

If  A  is  B,  Cis  D, 
but   A  is  By 
therefore,    C  is  Z>, 

the  conclusion  is  at  any  rate  apparently  obtained  by  a  com- 
bination of  two  premisses,  and  the  burden  of  proof  certainly 
seems  to  lie  with  those  who  deny  the  claims  of  such  an 
inference  as  this  to  be  called  mediate  inference. 

Professor  Bain's  arguments,  {Logic,  Deduction,-^.  117), 
upon  this  point  are  not  easy  to  formulate;  but  they  resolve 
themselves  into  one  or  other  or  both  of  the  following: — 

(i)  He  seems  to  argue  that  the  so-called  hypothetical 
syllogism  is  not  really  mediate  inference,  because  it  is  "a 
pure  instance  of  the  Law  of  Consistency";  in  other  words, 
because  "the  conclusion  is  implied  in  what  has  already  been 
stated."  But  is  not  this  the  case  in  all  formal  mediate 
inference?  Professor  Bain  cannot  consistently  maintain 
that  the  categorical  syllogism  is  more  than  a  pure  instance 


^  See  note  on  the  preceding  page- 


234 


SYLLOGISMS. 


[part  III. 


of  the  Law  of  Consistency;  or  that  the  conclusion  in  such  a 
syllogism  is  not  implied  in  what  has  already  been  stated. 

(2)  But  he  may  mean  that  the  conclusion  is  implied  in  the 
hypothetical  premiss  alone.  Indeed  he  goes  on  to  say,  "  '  If 
the  weather  continues  fine,  we  shall  go  into  the  country '  is 
transformable  into  the  equivalent  form  *The  weather  con- 
tinues fine,  and  so  we  shall  go  into  the  country.'  Any 
person  affirming  the  one,  does  not,  in  affirming  the  other, 
declare  a  new  fact,  but  the  same  fact."  If  this  is  intended 
to  be  understood  literally,  it  is  to  me  a  very  extraordinary 
statement.  Take  the  following  : — If  a  Russian  army  lands 
in  Britain,  the  volunteers  will  be  called  out ;  If  the  sun 
moves  round  the  earth,  modern  astronomy  is  utterly  wrong. 
Are  these  respectively  equivalent  to, — the  Russians  have 
landed  in  Britain  and  so  the  volunteers  are  being  called  out; 
the  sun  moves  round  the  earth,  and  so  modern  astronomy 
is  utterly  wrong  ?  Besides,  if  the  proposition  If  A  is  B, 
C  is  D  implies  that  A  is  B,  what  becomes  of  the  possible 
reasoning,  "  But  C  is  tiot  Z>,  therefore,  A  is  not  B"} 

Further  arguments  in  favour  of  Bain's  view  are  as 
follows  : — 

(i)  "There  is  no  middle  term  in  the  so-called  hypo- 
thetical syllogism."  The  answer  is  that  there  is  something 
in  the  premisses  which  does  not  appear  in  the  conclusion, 
and  that  this  corresponds  to  the  middle  term  of  the  cate- 
gorical syllogism.  If  we  reduce  the  hypothetical  syllogism 
to  the  categorical  form,  this  is  more  distinctly  recognisable. 

(2)  "  In  the  so-called  hypothetical  syllogism,  the  minor 
and  the  conclusion  indifferently  change  places."  This  state- 
ment is  erroneous.  Taking  the  syllogism  stated  at  the  com- 
mencement of  this  section  and  transposing  the  so-called 
minor  and  the  conclusion,  we  have  a  fallacy.  Compare 
section  203. 


CHAP.  VII.] 


SYLLOGISMS. 


235 


(3)  "The  major  in  a  so-called  hypothetical  syllogism 
consists  of  two  propositions,  the  categorical  major  of  two 
terms."  This  merely  tells  us  that  a  hypothetical  syllogism 
is  not  the  same  in  form  as  a  categorical  syllogism,  but 
seems  to  have  no  bearing  on  the  question  whether  the  so- 
called  hypothetical  syllogism  is  a  case  of  mediate  or  of 
immediate  inference. 

Turning  now  to  the  other  side  of  the  question,  I  do  not 
see  what  satisfactory  answers  can  be  given  to  the  following 
arguments  in  favour  of  regarding  the  hypothetico-categorical 
syllogism  as  a  case  of  mediate  inference.  In  any  such 
syllogism,  the  two  premisses  are  quite  distinct,  neither  can 
be  inferred  from  the  other,  but  both  are  necessary  in  order 
that  the  conclusion  may  be  obtained.  Again,  if  we  compare 
with  it  the  inferences  which  are  on  all  sides  admitted  to  be 
immediate  inferences  from  the  hypothetical  proposition,  the 
difference  between  the  two  cases  is  apparent.  From  If  A 
is  By  C  is  D  \  can  infer  immediately  If  C  is  not  Z>,  A  is 
not  B ;  but  I  require  also  to  know  that  C  is  not  D  in  order 
to  be  able  to  infer  that  A  is  not  B. 

It  has  also  been  shewn  that  a  reasoning  which  naturally 
falls  into  the  form  of  the  hypothetico-categorical  syllogism 
may  nevertheless  be  exhibited  in  the  form  of  the  ordinary 
categorical  syllogism,  which  is  admitted  to  be  a  case  of 
mediate  reasoning.  Moreover  there  are  distinct  forms, — 
the  inodus  ponens  and  the  modus  to/lens, — which  correspond 
to  distinct  forms  of  the  categorical  syllogism  ;  and  fallacies 
in  the  hypothetical  syllogism  correspond  exactly  to  certain 
fallacies  in  the  categorical  syllogism. 

Professor  Bowen  indeed  remarks  {logic,  p.  265): — "The 
reduction  of  a  Hypothetical  Judgment  to  a  Categorical 
shews  very  clearly  the  Immediacy  of  the  reasoning  in  what 
is  called  a  Hypothetical  Syllogism.    Thus,  If  A  is  B,  C  is  D, 


236 


SYLLOGISMS. 


[part  hi. 


is  equivalent  to  All  cases  of  ^  is  ^  are  cases  of  C  is  D, 
therefore, 

rSome  cases  of  ^  is  ^  are  cases  of  )  . 
(This  case  of  ^  is  ^  is  a  case  of  J  ^^^  ^^ 
But  does  not  this  overlook  the  fact  that  a  new  judgment  is 
required  to  tell  me  that  this  is  a  case  of  ^  is  ^  ?  The 
mere  statement  that  some  cases  oi  A  h  B  are  cases  of  C 
is  D  is  clearly  not  equivalent  to  the  conclusion  of  the 
hypothetical  syllogism. 

In  the  case  of  the  modiis  tollens, — "  If  ^  is  j9,  C  is  Z> ; 
but  C  is  not  D ;  therefore,  A  is  not  B  ",— the  argument  in 
favour  of  regarding  it  as  mediate  inference  is  still  more 
forcible ;  but  of  course  the  modus  ponens  and  the  modus 
tollens  stand  and  fall  together'. 

Professor  Croom  Robertson  {Mind,  1877,  P-  264)  has 
suggested  an  explanation  as  to  the  manner  in  which  this 
controversy  may  have  arisen.     He  distinguishes  the  hypo- 
thetical  "if"  from  the  inferential '' li,"  the  latter  being  equi- 
valent to  since,  seeing  that,  because.     No  doubt  by  the  aid  of 
a  certain  accentuation  the  word  "if"  may  be  made  to  carry 
with  it  this  force.     Professor  Robertson  quotes  a  passage 
from  Clarissa  Harlowe  in  which  the  remark  "  If  you  have 
the  value  for  my  cousin  that  you  say  you  have,  you  must 
needs  think  her  worthy  to  be  your  wife,"  is  explained  by  the 
speaker  to  mean,  ''Since  you  have,  &c."     Using  the  word 
in  this  sense,  the  conclusion  "C  is  Z>"  certainly  follows 
immediately  from  the  bare  statement,  "  If  AisB,  CisZ>"; 
or  rather  this  statement  itself  affirms  the  conclusion.     We 
cannot  however  regard  the  word  "if"  as  logically  carrying 
with  it  this  inferential  implication.     When  it  is  so  used  we 

*  In  section  i\o\  shew  further  that  the  Hypothetical  Syllogism  and 
the  Disjunctive  Syllogism  also  stand  and  fall  together. 


CHAP.  VII.]  SYLLOGISMS.  237 

have  really  a  condensed  mode  of  expression  including  two 
statements  in  one ;  I  should  indeed  turn  the  argument  the 
other  way  by  saying  that  in  the  single  statement  thus  in- 
terpreted we  have  a  hypothetical  syllogism  expressed 
elliptically  \ 

206.  If  A  is  true,  B  is  true ;  if  ^  is  true,  C  is 
true  ;  if  ^  is  true,  D  is  true.  What  is  the  effect  upon 
the  other  assertions  of  supposing  successively  (i)  that 
D  is  false ;  (2)  that  C  is  false ;  (3)  that  B  is  false ; 
(4)  that  A  is  false  ?  [Jevons,  Studies,  p.  146.] 

206.  Examine  the  following  : 

If  none  but  B  are  A,  it  cannot  be  possible  that 
any  X  are  F;  but  all  X  are  Y\  therefore  Some  A 
are  not  B, 

If  the  reasoning  is  correct,  reduce  it  to  proper 
syllogistic  mood  and  figure.  [v.] 

207.  Let  X,  F,  Z,  P,  Q,  R,  be  six  propositions  : 

given,  (a)  Of  X,  V,  Z,  one  and  only  one  is  true ; 

\b)  Of  P,  Qy  Ry  one  and  only  one  is  true  ; 

{c)  If  X  is  true,  P  is  true  ; 

{d)  If  F  is  true,  Q  is  true  ; 

{e)  If  Z  is  true,  R  is  true  ; 

prove  syllogistically, 

(/)  If  X  is  false,  Pis  false; 
{g)  If  F  is  false,  Q  is  false ; 
{li)    If  Z  is  false,  R  is  false. 


1  Cf.  Hansel's  Aldrich,  p.  103. 


CHAPTER  VIII. 

DISJUNCTIVE    SYLLOGISMS. 

208.     The  Disjunctive  Syllogism. 

A  Disjujictive  Syllogism  may  be  defined  as  a  formal 
reasoning  consisting  of  two  premisses  and  a  conclusion,  of 
which  one  premiss  is  disjunctive  while  the  other  premiss  and 
the  conclusion  are  categorical. 

For  example, 

A  is  either  B  or  C, 
A  is  not  B, 
therefore,   A  is  C. 

The  categorical  premiss  in  this  example  denies  one  of 
the  alternatives  stated  in  the  disjunctive  premiss,  and  we 

1  Archbishop  Thomson's  definition  of  the  disjunctive  syllogism— 
"  An  argument  in  which  there  is  a  disjunctive  judgment "  {Laws  of 
Thought,  p.  197)— must  I  think  be  regarded  as  too  wide.     It  would 
include  such  a  syllogism  as  the  following,— 

B  is  either  C  or  Z?, 
A  hB, 
therefore,         A  is  either  C  or  Z>. 

The  argument  here  in  no  way  turns  upon  the  disjunction,  and  the 
reasoning  may  be  regarded  as  an  ordinary  categorical  syllogism  in 
Barbara,  the  major  term  being  complex. 

A  more  general  treatment  of  reasonings  involving  disjunctive  judg- 
ments  is  given  in  Part  I  v. 


CHAP.  VIII.] 


SYLLOGISMS. 


239 


are  hence  enabled  to  affirm  the  other  alternative  as  our  con- 
clusion.    This  is  called  the  modus  tollendo  ponens. 

Some  logicians  also  recognise  as  valid  a  modus  ponendo 
tollens,  in  which  the  categorical  premiss  affirms  one  of  the 
alternatives  stated  in  the  disjunctive  premiss,  and  the  con- 
clusion denies  the  other  alternative.     Thus, 

A  is  either  B  or  (7, 
A  is  By 
therefore,  A  is  not  C. 
This  proceeds  on  the  assumption  that  the  elements  of 
the  disjunction  are  mutually  exclusive,  which  in  my  opinion 
is  not  necessarily  the  case'.     The  recognition  or  denial  of 
the  validity  of  the  modus  ponendo  tollens  depends  then  upon 
our  interpretation  of  the  disjunctive  proposition  itself 

209.  Comment  upon  the  following  definitions  of 
a  disjunctive  syllogism  : — 

"A  disjunctive  syllogism  is  a  syllogism  of  which 
the  major  premiss  is  a  disjunctive  and  the  minor  a 
simple  proposition,  the  latter  afilirming  or  denying  one 
of  the  alternatives  stated  in  the  former." 

"A  disjunctive  syllogism  is  a  syllogism  whose 
major  premiss  is  a  disjunctive  proposition." 

210.  Examine  the  question  whether  the  force  of 
a  Disjunctive  Proposition  as  a  premiss  in  an  argument 
is  equivalent  to  that  of  a  Hypothetical  Proposition. 

[L.] 

At  any  rate  so  far  as  the  disjunctive  syllogism  is  con- 
cerned this  question  must  be  answered  in  the  affirmative. 


1  Cf.  section  109. 


240  SYLLOGISMS.  [part  hi. 

A  is  either  B  or  C, 
A  is  not  B^ 
therefore,    A  is  C; 

may  be  resolved  into, — 

If  A  is  not  Bj  A  is  C, 
A  is  not  B, 
therefore,    A  i%  C) 

or,  into, — 

If  A  is  not  Q  A  is  B, 
A  is  not  B, 
therefore,  A  is  C. 

It  may  be  observed  that  those  who  deny  the  character  of 
mediate  reasoning  to  the  hypothetical  syllogism  must  also 
deny  it  to  the  disjunctive  syllogism,  or  else  they  must  refuse 
to  recognise  the  resolution  of  the  disjunctive  proposition 
into  one  or  more  hypothetical  propositions. 

211.  Is  it  possible  to  apply  distinctions  of  Figure 
either  to  Hypothetical  or  to  Disjunctive  Syllogisms  ? 

[C] 

212.  Comment  upon  Jevons's  statement: — "It 
will  be  noticed  that  the  disjunctive  syllogism  is 
governed  by  totally  different  rules  from  the  ordinary 
categorical  syllogism,  since  a  negative  premiss  gives 
an  affirmative  conclusion  in  the  former,  and  a  negative 
in  the  latter/' 

213.  If  all  things  are  either  X  or  F,  and  all  things 
are  either  Y  or  Z,  what  inference  can  you  draw  > 

[Jevons,  Studies,  p.  303.] 


CHAP.  VIII.]  SYLLOGISMS. 

214.     The  Dilemma. 


241 


The  proper  place  of  the  Dilemma  among  Conditional 
Arguments  is  made  puzzling  by  the  fact  that  conflicting 
definitions  of  the  Dilemma  are  given  by  different  logical 
writers.  It  will  be  useful  to  comment  briefly  upon  some  of 
these  definitions. 

(i)  Mansel  {Aldrich^  p.  108)  defines  the  Dilemma  as 
"a  syllogism  having  a  conditional  (hypothetical)  major 
premiss  7vith  more  than  07te  antecedent^  and  a  disjunc- 
tive minor."  Equivalent  definitions  are  given  by  Whately 
and  Jevons. 

Three  forms  of  dilemma  are  recognised  by  these 
writers : — 


1. 


The  Simple  Constructive  Dilemma. 

If  ^  is  -5,  C  is  Z> ;  and  if  ^  is  i^,  C'\s  D  \ 
But  either  ^  is  ^  or  -£^  is  F\ 
Therefore,  C  is  D. 


11 


The  Complex  Constructive  Dilemma. 

If  ^  is  ^,  C  is  Z> ;  and  if  ^  is  i^  6^  is  H\ 
But  either  ^  is  ^  or  ^  is  F-, 
Therefore,  Either  C  is  Z?  or  Gisif, 

iii.  The  Destructive  Dilemma,  (always  Complex), 

U  AisB,  CisZ>;  and  if  E  is  F,  G  is  H; 
But  either  C  is  not  Z>  or  6^  is  not  H', 
Therefore,  Either  A  is  not  B  01  E  i^  not  A 

The  Destructive  Dilemma  is  said  to  be  always  complex; 
and  the  simple  form  corresponding  to  the  third  of  the 
above  is  certainly  excluded  by  the  definition  given.  It  would 
run, — 

K.  L.  16 


242 


SYLLOGISMS. 


[part  iir. 


U  AisB,  CisZ>;  and  if  A  is  M,  E  is  E; 
But  either  C  is  not  Z>  or  i^  is  not  F; 
Therefore,  A  is  not  B  ; 

and  here  there  is  onfy  one  aniecederit  in  the  major. 

But  the  question  arises  whether  such  exclusion  is  not 
arbitrary,  and  whether  this  definition  ought  not  therefore  to 
be  rejected. 

Whately  regards  the  name  Dilemma  as  necessarily  im- 
plying two  antecedents ;  but  does  it  not  rather  imply  two 
alternatives^  each  of  which  is  equally  distasteful?  He 
goes  on  to  assert  that  the  excluded  form  is  merely  a  de- 
structive hypothetical  syllogism,  similar  to  the  following, — 

U  AhJB,  Cis  D; 
C  is  not  D ; 
therefore,  A  is  not  B. 

But  the  two  really  differ  precisely  as  the  simple  constructive 
dilemma, — 

If  A  is  B,  Cis  £>;  and  if  i^:  is  F,  C is  £> ; 
But  either  y^  is  -^  or  ^  is  E; 
therefore,  C  is  £>; — 

differs  from  the  constructive  hypothetical  syllogism, — 

If  A  is  B,  Cis  D  : 
AisB; 
therefore,  C  is  Z>. 

Besides,  it  is  clear  that  it  is  not  merely  a  destructive  hypo- 
thetical syllogism  such  as  has  been  already  discussed,  since 
the  premiss  which  is  combined  with  the  hypothetical  premiss 
is  not  categorical  but  disjunctive '. 

^  The  argument, — 

If  AhB,  Cis  D  and  £  is  F; 

But  either  C  is  not  Z>  or  i?  is  not  F; 

Therefore,  A  is  not  B ; 


CHAP.  VIII.] 


SYLLOGISMS. 


243 


(2)  Professor  Fowler  {Deductive  Logic,  p.  116)  gives 
the  following : — "There  remains  the  case  in  which  one 
premiss  of  the  complex  syllogism  is  a  conjunctive,  (i.e.,  a 
hypothetical),  and  the  other  a  disjunctive  proposition,  it 
being  of  course  understood  that  the  disjunctive  proposition 
deals  only  with  expressions  which  have  already  occurred  in 
the  conjunctive  proposition.     This  is  called  a  Dilemma^ 

Under  this  definition,  it  is  no  longer  required  that  there 
shall  be  at  least  two  antecedents  in  the  hypothetical  pre- 
miss ;  and  hence,  four  forms  are  included,  namely,  the  two 
constructive  dilemmas,  and  a  simple  as  well  as  a  complex 
destructive. 

(3)  The  following  definition  is  sometimes  given: — 
**The  Dilemma  (or  Trilemma  or  Polylemma)  is  a  syllogism 
in  which  two  (or  three  or  more)  alternatives  are  given  in 
one  premiss,  but  in  the  other  it  is  shewn  that  in  any  case 
the  same  conclusion  follows." 

This  would  include  the  simple  constructive  dilemma  and 
the  simple  destructive  dilemma,  (as  already  given);  but  it 
would  not  allow  that  either  of  the  complex  dilemmas  is 


must  be  distinguished  from  the  following, — 

If  ^is^,  CisZ>and£isi^; 
But  C  is  not  D,  and  E  is  not  F\ 
Therefore,  A  is  not  B, 

In  the  latter  of  these  there  is  no  alternative  given  at  all,  and  the 
reasoning  is  equivalent  to  two  simple  hypothetical  syllogisms,  yielding 
the  same  conclusion,  namely, — 

(i)  If  ^  is^,  CisZ>; 
But  C  is  not  D ; 
Therefore,  A  is  not  B. 
(2)  If  A  is  By  E  is  F', 
But  E  is  not  F\ 
Therefore,  A  is  not  B. 

16 — 2 


244 


SYLLOGISMS. 


[part  III. 


properly  so-called,  since  in  each  case  we  are  left  with  the 
same  number  of  alternatives  in  the  conclusion  as  are  con- 
tained in  the  disjunctive  premiss. 

This  definition,  however,  embraces  forms  that  are  ex- 
cluded by  both  the  preceding  definitions.     For  example, 

If  A  is,  either  B  ox  C  is ; 

But   neither  B  nor  C  is  ; 

Therefore,  A  is  not\ 

(4)  Hamilton  {Logic,  i.  p.  350)  defines  the  Dilemma 
as  "  a  syllogism  in  which  the  sumption  (major)  is  at  once 
hypothetical  and  disjunctive,  and  the  subsumption  (minor) 
sublates  the  whole  disjunction,  as  a  consequent,  so  that  the 
antecedent  is  sublated  in  the  conclusion."  This  involved 
definition  appears  to  have  chiefly  in  view  the  form  last  given, 
namely, — 

If  A  is,  either  ^  is  or  (7  is  ; 
Neither  B  is  nor  C  is ; 
Therefore,  A  is  not ; 

but  it  excludes  the  following, — 

If  A  is,  C  is  j  and  if  B  is,  C  is ; 
But  either  A  is  or  ^  is ; 
Therefore,   C  is. 

This  however  is  one  of  the  typical  forms  of  Dilemma 
according  to  all  the  preceding  definitions. 

(5)  Thomson  {Laics  of  Thought,  p.  203)  gives  the  follow- 
ing,—"A  dilemma  is  a  syllogism  with  a  conditional  (hyiDo- 
thetical)  premiss,  in  which  either  the  antecedent  or  the  con- 
sequent is  disjunctive." 

This  definition  is  probably  wider  than  Thomson  himself 
intended.     It  would  include  such  forms  as  the  following : 

*  Cf.  Ucberweg,  System  of  Logic,  Lindsay's  translation,  p.  457. 


245 


CHAP.  VIII.]  SYLLOGISMS. 

liAh^B  ox  E  is  F,  then  C  is  D; 

But  C  is  not  D ; 

Therefore,  A  is  not  B,  and  E  is  not  F. 

If  A'l^B,  C  hn  orE  is  F; 

But  A  is  B; 

Therefore,  C  is  L>  or  E  is  F. 

215.  "  Dilemmatic  arguments  are  more  often  fal- 
lacious than  not."     Why  is  this  ?  [c] 

Jevons  {E/emenfs  of  Logic,  p.  168)  remarks  that  "Dilem- 
matic arguments  are  more  often  fallacious  than  not,  because 
it  is  seldom  possible  to  find  instances  where  two  alternatives 
exhaust  all  the  possible  cases,  unless  indeed  one  of  them  be 
the  simple  negative  of  the  other."  In  other  words,  most 
dilemmatic  arguments  will  be  found  to  contain  a  false 
premiss.  It  is  however  somewhat  misleading  to  say  that  a 
syllogistic  argument  is  fallacious  because  it  contains  a  false 
premiss.  At  any  rate,  notwithstanding  this,  the  argument 
itself  from  the  point  of  view  of  Formal  Logic  may  be  per- 
fectly cogent. 

216.  What  can  be  inferred  from  the  premisses, 
Either  AxsBox  C\s  Z>,  Either  C  is  not  DoxE  is  not  F ? 
Exhibit  the  reasoning  in  the  form  of  a  dilemma. 


CHAPTER  IX. 


THE    QUANTIFICATION    OF   THE    PREDICATE. 

217.  The  eight  propositional  forms  resulting 
from  the  explicit  Quantification  of  the  Predicate. 

The  fundamental  postulate  of  Logic,  according  to  Sir  W. 
Hamilton,  was  "that  we  be  allowed  to  state  explicitly  in 
language  all  that  is  implicitly  contained  in  thought";  and 
since  he  also  maintained  that  "in  thought  the  predicate  is 
always  quantified,"  he  made  it  follow  immediately  from  his 
postulate,  that  "in  logic,  the  quantity  of  the  predicate  must 
be  expressed,  on  demand,  in  language." 

This  doctrine  of  the  explicit  quantification  of  the  predi- 
cate led  Hamilton  to  recognise  eight  distinct  propositional 
forms  instead  of  the  customary  four : — 

All  5  is  all  P,  U. 

All  S  is  some  P,  A. 

Some  6"  is  all  /*,  Y, 

Some  S  is  some  P,  1. 

No  S  is  any  P,  E, 

No  S  is  some  /*,  ff. 

Some  ^  is  not  any  F,        O. 
Some  S  is  not  some  P.      w. 
The  symbols  here  attached  are  due  to  Thomson  ^  and 
they  are  the  ones  in  most  common  use. 

1  Thomson  however  rejects  the  fornis  rj  and  ta. 


CHAP.  IX.] 


SYLLOGISMS. 


247 


The  symbols  used  by  Hamilton  himself  were  A/a,  Afi, 
//a,  Ifi,  Alia,  Ani,  Ina,  Int.  Here/  indicates  an  affirmative 
proposition,  n  indicates  a  negative;  a  means  that  the  cor- 
responding term  is  distributed,  i  that  it  is  undistributed. 

Spalding's  symbols  [Logic,  p.  83)  are  A^,  A,  P,  I,  E, 
\E,  O,  \0.  Mr  Carveth  Read  (Theory  of  Logic,  p.  193) 
suggests  A\  A,  r,  I,  E,  E^,  O,  O^, 

The  equivalence  of  these  various  symbols  is  shewn  in 
the  following  table  : — 


Thomson. 

Hamilton. 
Afa 

! 

Spalding. 

Carveth  Read. 

All  ^  is  all /^ 

U 
A 

Y 

1 

A^ 

\ 

A^ 

All  S  is  some  P 

Afi 

1 
A 

A 

Some  S  is  all  P 

Ifa 

I 

E 

72 

Some  S  is  some  P 

I 

r-                        -  - 

E 

! — 

0 

Ifi 
Ana 
Ani 
Ina 
In?- 

/ 

1 

. * 

No  S  is  any  P 

E 

No  S  is  some  P 

\E 

E. 

Some  6"  is  not  any  P 

0 

0 

Some  S  is  not  some  P 

1 

iO 

0^ 

218.  The  meaning  to  be  attached  to  the  word 
some  in  the  eight  propositional  forms  recognised  by 
Sir  William  Hamilton. 

Professor  Baynes,  in  his  authorised  exposition  of  Sir 
William  Hamilton's  new  doctrine,  would  at  the  outset  lead 


248 


SYLLOGISMS. 


[part  III. 


one  to  suppose  that  we  have  no  longer  to  do  with  the  in- 
determinate ''some"  of  the  Aristotelian  Logic,  but  that  this 
word  is  now  to  be  used  in  the  more  definite  sense  of  ''''some, 
hilt  7iot  all.'"     We  have  seen  that  the  fundamental  postulate 
of  Logic  on  which  Hamilton  bases  his  doctrine  is  "  that  we 
be  allowed  to  state  explicitly  in  language,  all  that  is  im- 
plicitly contained  in  thought " ;  and  applying  this  postulate, 
Mr  Baynes  {Ne%v  A?ialytic  of  Logical  Fo7'ms)  remarks : — 
''Predication  is   nothing  more  or  less  than  the  expression 
of  the  relation  of  quantity  in  which  a  notion  stands  to  an 
individual,  or  two  notions  to  each  other.     If  this  relation 
were  indeterminate — if  we  were  uncertain  whether  it  was 
of  part,  or  whole,  or  none — there  could  be  no  predication. 
Since,   therefore,   the   predicate    is    always    quantified    in 
thought,  the  postulate  applies;  i.e.^  in  logic,  the  quantity  of 
the  predicate  must  be  expressed,  on  demand,  in  language. 
For  example,  if  the  objects  comprised  under  the  subject 
be  some  part,  but  not  the  whole,  of  those  comprised  under 
the  predicate,  we  write  All  X  is  some  F,  and  similarly  with 
other  forms." 

But  if  it  is  true  that  we  know  definitely  the  relative 
extent  of  subject  and  predicate,  and  if  "some"  is  used 
strictly  in  the  sense  of  "  some  but  not  all,"  we  should  have 
hw'^five  propositional  forms  instead  of  eight,  namely, — All  S 
is  all  Py  All  S  is  some  Py  Some  S  is  all  P,  Some  S  is  some  P\ 
No  S  is  any  P. 

We  have  already  shewn  (section  95)  that  the  only 
possible  relations  between  two  terms  in  respect  to  their 
extension  are  given  by  the  five  diagrams, — 


^  Using  sotne  in  the  sense  here  indicated,  Some  S  is  some  P  neces- 
sarily implies  Some  S  is  not  any  P  and  No  S  is  some  P. 


CHAP.  IX.] 


SYLLOGLSMS. 


249 


0, 


These  correspond  respectively  to  the  above  five  propo- 
sitions; and  it  is  clear  that  on  the  view  indicated  by 
Mr  Baynes  the  eight  forms  are  redundant.  This  point  is 
worked  out  in  detail  by  Mr  Venn  {Symbolic  Logic,  Chap,  i.); 
he  shews  the  utter  inadequacy  and  unscientific  character  of 
the  Hamiltonian  doctrine. 

I  am  altogether  doubtful  whether  writers  who  have 
adopted  the  eightfold  scheme  have  themselves  recognised 
the  pitfalls  that  surround  the  use  of  the  word  sojnc.  Many 
passages  might  be  quoted  in  which  they  distinctly  adopt  the 
meaning — "  some,  not  all."  Thus,  Thomson  {Laws  of 
llioiighty  p.  150)  makes  U  and  A  inconsistent.  Bowen 
{LogiCy  pp.  169,  170)  would  pass  from  I  to  O  by  imme- 
diate inference'.  Hamilton  himself  would  agree  with 
Thomson  and  Bowen  on  these  points ;  but  he  is  curiously 
indecisive  on  the  general  question  here  raised.  He  remarks 
{LogiCy  II.  p.  282)  that  some  "is  held  to  be  a  definite  some 
when  the  other  term  is  definite,"  /.  c.y  in  A  and  Y,  r/  and 
O  ;  but  "on  the  other  hand,  when  both  terms  are  indefinite 
or  particular  the  some  of  each  is  left   wholly  indefinite," 

1  "  This  sort  of  Inference,"  he  says,  "  Hamilton  would  call  Inte- 
gration, as  its  effect  is,  after  determining  one  part,  to  reconstitute  the 
whole  by  bringing  into  view  the  remaining  part." 


250 


SYLLOGISMS. 


[PART  III.  I  CHAP.  IX.] 


SYLLOGISMS. 


251 


/>.,  in  I  and  a>^  This  is  very  confusing,  and  it  would  be 
most  difficult  to  apply  the  distinction  consistently.  Hamil- 
ton himself  certainly  does  not  so  apply  it.  For  example,  on 
his  view  it  should  no  longer  be  the  case  that  two  affirmative 
premisses  necessitate  an  affirmative  conclusion;  nor  that 
two  negative  premisses  invalidate  a  syllogism.  Thus,  the 
following  should  be  regarded  as  valid : — 

All  P  is  some  J/, 
All  M  is  some  S, 

therefore,  Some  S  is  not  any  P. 

No  M  is  any  P^ 
Some  S  is  not  any  M^ 

therefore,  Some  S  is  not  any  P. 

Such  syllogisms  as  these,  however,  are  not  admitted  by 
Hamilton  and  Thomson.  Hamilton's  supreme  canon  of 
the  categorical  syllogism  {Logic,  11.  p.  357)  is  : — "  What 
worse  relation  of  subject  and  predicate  subsists  between 
either  of  two  terms  and  a  common  third  term,  with  which 
one,  at  least,  is  positively  related ;  that  relation  subsists 
between  the  two  terms  themselves."     This  clearly  provides 

^  Mr  Lindsay,  however,  in  expounding  Hamilton's  doctrine  {Ap- 
pcfidix  to  Uebcnoe^s  System  of  Logic,  p.  580)  says  more  decisively, — 
"  Since  the  subject  must  be  equal  to  the  predicate,  vagueness  in  the 
predesignations  must  be  as  far  as  possible  removed.  Some  is  taken 
as  equivalent  to  some  but  not  all''' 

Spalding  (Logic,  p.  184)  definitely  chooses  the  other  alternative. 
He  remarks  that  in  his  own  treatise  "the  received  interpretation 
so7ne  at  least  is  steadily  adhered  to." 

Mr  Carveth  Read  [Theory  of  Logic,  p.  196)  distinguishes  two 
schemes  of  what  he  calls  Bidesignate  Relationships  (Quantified  Pre- 
dicates) in  one  of  which  the  sign  Sotne  is  understood  to  mean  Some 
only,  and  in  the  other  Some  at  least.  In  each  case,  however,  he  seems 
to  retain  eight  distinct  propositional  forms. 


I 


i 


\ 


1 

i 


that  one  premiss  at  least  shall  be  affirmative,  and  that  an 
affirmative  conclusion  should  follow  from  two  affirmative 
premisses.  Thomson  {La7ifs  of  Thought,  p.  165)  explicitly 
lays  down  the  same  rules.  Here  then  is  further  evidence  of 
the  unscientific  nature  of  the  Hamiltonian  doctrine.  The 
same  subject  is  pursued  further  in  the  three  following 
sections. 

219.  What  results  would  follow  if  we  were  to 
interpret  '  Some  ^'s  are  ^'s '  as  implying  that  '  Some 
other  As  are  not  ^'s '  } 

[Jevons,  Studies  in  Deductive  Logic,  p.  15 1.] 

Professor  Jevons  himself  answers  this  question  by  say- 
ing, ''The  proposition  'Some  A\  are  ^'s'  is  in  the  form  I, 
and  according  to  the  table  of  opposition  I  is  true  if  A  is 
true ;  but  A  is  the  contradictory  of  O,  which  would  be  the 
form  of  'Some  other  v4's  are  not  ^'s.'  Under  such  cir- 
cumstances A  could  never  be  true  at  all,  because  its  truth 
would  involve  the  truth  of  its  own  contradictory,  which  is 

absurd." 

This  is  turning  the  criticism  the  wrong  way,  and  proves 
too  much.  It  is  not  true  that  we  necessarily  involve  our- 
selves in  self-contradiction  if  we  use  some  in  the  sense 
of  some  only.  What  should  be  pointed  out  is  that  if  we 
use  the  word  in  this  sense,  the  truth  of  I  no  longer  follows 
from  the  truth  of  A ;  but  on  the  other  hand  these  two  pro- 
positions are  inconsistent  with  each  other. 

Taking  the  five  propositional  forms  which  are  obtained 
by  attaching  this  meaning  to  some,  namely, — All  S  is  all 
P,  All  S  is  some  P,  Some  S  is  all  P,  Some  S  is  some  P, 
No  S  is  P,—\\.  should  be  obser\^ed  that  each  one  of  these 
propositions  is  inconsistent  with  each  of  the  others,  and 
also  that  no  one  is  the  contradictory  of  any  one  of  the 


252 


SYLLOGISMS. 


[part  III. 


CHAP.  IX.] 


others.  If,  for  example,  on  this  scheme  we  ^vish  to  ex- 
press the  contradictory  of  U,  we  can  only  do  so  by  affirm- 
ing an  alternative  between  Y,  A,  I  and  E. 

Nothing  of  all  this  appears  to  have  been  noted  by  the 
Hamiltonian  writers  \  even  in  the  cases  in  which  they  ex- 
plicitly profess  to  use  S(?we  in  the  sense  of  "  some  oniyr 

How  the  above  five  forms  may  be  expressed  by  means 
of  the  ordinary  Aristotelian  four  forms  has  been  discussed 
in  section  99. 

220.  If  in  the  eight  Hamiltonian  forms  of  pro- 
position some  is  used  in  the  ordinary  logical  sense, 
what  is  the  precise  information  given  by  each  of 
these  propositions } 

laking   the  five  possible  relations  between  two  terms, 
and  numbering  them  as  follows, — 

(^)  (2)  (3) 


we  may  write  against  each  of  the  propositional  forms  the 
relations  which  are  compatible  ^vith  it^: — 

1  Thomson  {Laws  of  Thought,  p.  149)  gives  a  scheme  of  oppo- 
sition in  which  I  and  E  appear  as  contradictories,  l)ut  A  and  0  as 
contraries.  He  appears  to  use  some  in  the  sense  of  sovie  hut  not  all 
in  the  case  of  A  and  Y  only. 

^  If  the  Hamiltonian  writers  had  attempted  to  illustrate  their  doc- 


•f-r. 


-"P 


M 

•1  ' 


SYLLOGISMS. 

u 

A 

I 

I,   2 

Y 

I.  3 

,„., 

I 
E 

V 

I.  2,  3,  4 

5 

2,  4,  5 

0 

3,  4,  5 

io 

i»  2,  3,  4,  5 

253 


We  have  then  the  following  pairs  of  contradictories,— 
A,  O;  Y,  17;  I,  E.  The  contradictory  of  U  is  obtained 
by  affirming  an  alternative  between  rj  and  O. 

We   may  point  out  how  each  of  the  above  would   be 
expressed  without  the  use  of  quantified  predicates  : — 

U  =  Sal",  PaS; 
A=SajP; 
Y  =  FaS; 

l=SiP- 
K^SeP; 

V=FoS; 
O  =  SoP. 

trine  by  means  of  the  Eulerian  diagrams,  they  v^ould  I  think  either 
have  found  it  to  be  unworkalde,  or  they  would  have  worked  it  out  to 
a  more  distinct  and  consistent  issue. 


254 


SYLLOGISMS. 


[part  111. 


What  exact  information,  if  any,  is  given  by  w   is   dis- 
cussed in  the  following  section. 

221.     The  Hamiltonian  proposition  o),  "  Some  5 
is  not  some  PT 

The  proposition  w,  "Some  S  is   not  some  /^,"  is  not 
inconsistent  with  any  of  the  other  prepositional  forms,  not 
even  with  U,  "All  S  is  all  Pr     For  example,  "all  equi- 
lateral triangles  are  all  equiangular  triangles,"  yet  never- 
theless   "this   equilateral  triangle    is    not    that    equilateral 
triangle,"  which  is  all  that  w  asserts.     "Some  S  is  some 
P''  is  indeed  always    true    except  when   both  the  subject 
and  the  predicate  are  the  name  of  an  individual  and  the 
same  individual.     De  Morgan'  {Syllabus,  p.  24)  points  out 
that  its  contradictory  is,-—"  5  and  /^  are  singular  and  identi- 
cal ;  there  is  but  one  S,  there  is  but  one  P,  and  S  is  P." 
It  may  be   said  without  hesitation  that  the  proposition  <i> 
is  of  absolutely  no  logical  importance. 

222.  To  what  extent  do  the  eight  forms  result- 
ing from  predicating  of  all  or  some  trains,  that  they 
do  or  do  not,  stop  at  all  or  some  stations,  coincide  in 
significance  with  Hamilton's  schedule }  In  particular, 
do  the  objections  to  "  Some  A  is  not  some  i?"  apply 
to  the  proposition  "  Some  trains  do  not  stop  at  some 
stations  " }  ry-j 

223.  Examine  Thomson's  statement  that  "9;  has 
the  semblance  only,  and  not  the  power  of  a  denial. 
True  though  it  is,  it  does  not  prevent  our  making 
another  judgment  of  the  affirmative  kind,  from  the 
same  terms." 

^  De  Morgan  in  several   passages   criticizes   with  great  acuteness 
the  Hamiltonian  scheme  of  propositions. 


CHAP.  IX.] 


SYLLOGISMS. 


255 

224.  Write  out  the  various  judgments,  including 
U  and  Y,  which  are  logically  opposed  to  the  judg- 
ment :  No  puns  are  admissible.  State  in  the  case  of 
each  judgment  thus  formed  what  is  the  kind  of  op- 
position in  which  it  stands  to  the  original  judgment, 
and  also  the  kind  of  opposition  between  each  pair  of 
the  new  judgments.  [c] 

225.  Explain  precisely  how  it  is  that  O  admits 
of  ordinary  conversion  if  the  principle  of  the  Quanti- 
fication of  the  Predicate  is  adopted,  although  not 
otherwise. 

226.  Test  the  validity  of  the  following  syllogisms, 
and  examine  whether  or  not  the  reasoning  contained 
in  those  that  are  valid  can  be  expressed  without  the 
use  of  quantified  predicates  : — 

In  Figure  i,  UUU  IUt;. 

In  Figure  2,  7?UO. 

In  Figure  3,  YAY,  Yt/E. 

(i)    UUU  in  Figure  i  is  valid:  - 

All  Mis  all  P, 

All  S  is  all  M, 

therefore,  All  S  is  all  P. 

It  should  be  noticed  that  whenever  one  of  the  premisses 
is  U,  the  conclusion  may  be  obtained  by  substituting  S  or 
P  (as  the  case  may  be)  for  M  in  the  other  premiss. 

Without  the  use  of  quantified  predicates,  the  above 
reasoning  may  be  expressed  by  means  of  the  two  syllo- 
gisms,— 

AllMisP,  AllMisS, 

All  S  is  M,  All  P  is  Af, 

therefore,  All  S  is  P.  therefore,  All  P  is  S. 


256 


SYLLOGISMS. 


[part  III. 


(2)  lU rj  in  Figure  i  is  invalid,  if  scwie  is  used  in  its 
ordinary  logical  sense.  The  premisses  are  So;/ie  M  is  some 
P,  and  A//  S  is  all  M.  We  may  therefore  obtain  the 
legitimate  conclusion  by  substituting  S  for  AT  in  the  major 
premiss.     This  yields  Some  S  is  some  P, 

If,  however,  so7ne  is  here  used  in  the  sense  of  soine  only. 
No  S  is  some  P  follows  from  some  S  is  some  P^  and  the 
original  syllogism  is  valid,  although  a  negative  conclusion  is 
obtained  from  two  affirmative  premisses. 

This   syllogism  is   given   valid  by  Thomson  {Laws  of 
T/umg/it,  p.  188);  but  apparently  only  through  a  misprint 
for  IE17.     Using  so?fie  in  the  sense  of  so7ne  07ily\  several 
other  syllogisms  would  be  valid  that  he  does  not  give  as 
such\ 

(3)  T/UO  in  Figure  2  is  valid: — 

No  P  is  some  M, 
All  S  is  all  M, 
therefore.  Some  S  is  tiot  any  P. 

Without  the  use  of  quantified  predicates,  we  can  obtain 
an  equivalent  argument  in  Bocardo,  thus, — 

Some  M  is  7iot  J\ 
All  M  is  S, 
therefore,  So7ne  S  is  7iot  P. 

(4)  Y A Y  in  Figure  3  is  valid : — 

So77ie  M  is  all  P, 
All  M  is  so77ie  S, 
therefore,  So77ie  S  is  all  P. 

AV'ithout   quantified    predicates    the    reasoning  may   be 
expressed  in  Parl^a/a,  thus, — 


*■  Cf.  section  218. 


CHAP.  IX.] 


SYLLOGISMS. 


257 


All  M  is  S, 
All  P  is  M, 
therefore.  All  P  is  S. 

(5)   YrjK  in  Figure  3  is  invalid  : — 

From  So77te  M  is  all  P, 

and  No  M  is  so77ie  S, 

we  infer  that  No  S  is  a7iy  P ; 

but  this  involves  illicit  process  of  the  minor. 

227.  Examine  the  validity  of  the  following 
moods : — 

In  Figure  i,  UAU,  YOO,  EYO  ; 
In  Figure  2,  AAA,  AYY,  UOo); 
In  Figure  3,  YEE,  OYO,  Aa)0.  [c] 

228.  In  what  figures,  if  any,  are  the  following 
moods  valid  }  Where  the  conclusion  is  weakened, 
point  out  the  fact : — 

AUI;  YAY;  UO77;   IU7;;   UEO.  [l.] 

229.  Is  it  possible  that  there  should  be  three 
propositions  such  that  each  in  turn  is  deducible  from 
the  other  two  }  [v.] 

230.  The  Figured  and  the  Unfigured  Syllogism. 

The  distinction  between  the  figured  and  the  unfigured 
syllogism  is  due  to  Hamilton,  and  is  connected  with  his 
doctrine  of  the  Quantification  of  the  Predicate. 

By  a  rigid  quantification  of  the  predicate  the  distinction 
between  subject  and  predicate  may  be  dispensed  with ;  and 
such  being  the  case  there  is  no  ground  left  for  distinction 
of  figure,  (which  depends  upon  the  position  of  the  middle 
term   as   subject  or    predicate   in   the  premisses).      This 

K.  L.  17 


258 


SYLLOGISMS. 


[part  III. 


gives  what  Hamilton  calls  the   Unfgured  Syllogism.     For 

example, — 

Any  bashfulness  and  any  praiseworthy  are  not  equivalent, 
All  modesty  and  some  praiseworthy  are  equivalent, 

therefore.  Any  bashfulness  and  any  modesty  are  not  equi- 
valent. 

All  whales  and  some  mammals  are  equal, 

All  whales  and  some  water  animals  are  equal, 
therefore.  Some  mammals  and   some  water   animals  are 
equal. 

There  is  an  approach  here  towards  the  Equational  Logic. 

Hamilton  gives  a  distinct  canon  for  the  unfigured 
syllogism  as  follows : — **  In  as  far  as  two  notions  either 
both  agree,  or  one  agreeing  the  other  does  not,  with  a 
common  third  notion ;  in  so  far  these  notions  do  or  do  not 
agree  with  each  other." 


CHAPTER  X. 


EXAMPLES   OF   ARGUMENTS   AND   FALLACIES. 


231.  Examine  technically  the  following  argu- 
ments : — 

(i)  Those  who  hold  that  the  Insane  should  not 
be  punished  ought  in  consistency  to  admit  also  that 
they  should  not  be  threatened ;  for  it  is  clearly  un- 
just to  punish  any  one  without  previously  threaten- 
ing him. 

(2)  If  he  pleads  that  he  did  not  steal  the  goods, 
why,  I  ask,  did  he  hide  them,  as  no  thief  ever  fails 
to  do .?  [v.] 

232.  Examine  technically  the  following  argu- 
ments : — 

Knavery  and  folly  always  go  together ;  so,  know- 
ing him  to  be  a  fool  I  distrusted  him. 

If  I  deny  that  poverty  and  virtue  are  inconsistent, 
and  you  deny  that  they  are  inseparable,  we  can  at 
least  agree  that  some  poor  are  virtuous. 

How  can  you  deny  that  the  infliction  of  pain  is 
justifiable  if  punishment  is  sometimes  justifiable  and 
yet  always  involves  pain  }  '  [V.J 

17 — 2 


26o  SYLLOGISMS.  [part  hi. 

233.  Test  the  following  : — 

"If  all  men  were  capable  of  perfection,  some 
would  have  attained  it;  but,  none  having  done  so, 
none  are  capable  of  it."  [v.] 

234.  Examine  the  following  reasoning : — 

How  can  you  deny  that  any  poor  should  be  re- 
lieved, when  you  deny  that  sickness  and  poverty  arc 
inseparable,  and  also  that  any  sick  should  not  be 
relieved  ?  [v.] 

235.  In  how  many  different  syllogistic  moods 
could  you  express  the  reasoning  in  the  following 
sentence  by  supplying  the  proper  premisses  ? 

These  plants  cannot  be  orchids,  for  they  have 
opposite  leaves.  [v.] 

236.  In  how  many  different  moods  may  the 
argument  implied  in  the  following  proposition  be 
stated  ? 

"No  one  can  maintain  that  all  persecution  is 
justifiable  who  admits  that  persecution  is  sometimes 
ineffective." 

How  would  the  formal  correctness  of  the  reason- 
ing be  affected  by  reading  "deny"  for  "maintain"  ? 

[v.] 

237.  What  conclusions  (if  any)  can  be  drawn 
from  each  pair  of  the  following  sentences  taken  two 
and  two  together  ? 

(i)  None  but  gentlemen  are  members  of  the 
club ; 


CHAP.  X.]  SYLLOGISMS.  261 

(2)  Some  members  of  the  club  are  not  officers  ; 

(3)  All  members  of  the  club  are  invited  to 
compete ; 

(4)  All  officers  are  invited  to  compete. 

Point  out  the  mood  and  figure  in  each  case  in 
which  you  make  a  valid  syllogism ;  and  state  your 
reasons  when  you  consider  that  no  valid  syllogism  is 
possible.  [v.] 

238.  "  No  wise  man  is  unhappy ;  for  no  dishonest 
man  is  wise,  and  no  honest  man  is  unhappy." 

Examine  this  inference,  and  if  you  think  it  sound 
resolve  it  into  a  regular  syllogism.  [w.] 

239.  Detect  the  fallacy  in  the  following  argu- 
ment : — 

"  A  vacuum  is  impossible,  for  if  there  is  nothing 
between  two  bodies  they  must  touch."  [n.] 

240.  Write  the  following  arguments  in  syllogistic 
form,  and  reduce  them  to  Figure  i  : — 

(a)  Falkland  was  a  royalist  and  a  patriot ;  there- 
fore, some  royalists  were  patriots. 

(^)  All  who  are  punished  should  be  responsible 
for  their  actions ;  therefore,  if  some  lunatics  are  not 
responsible  for  their  actions,  they  should  not  be 
punished. 

(7)  All  who  have  passed  the  Little-Go  have  a 
knowledge  of  Greek  ;  hence  A.  B.  cannot  have  passed 
the  Little-Go,  for  he  has  no  knowledge  of  Greek. 


262 


SYLLOGISMS. 


[part  hi. 


241.  Whately  says, — "  *  Every  true  patriot  is  dis- 
interested, few  men  are  disinterested,  therefore  few 
men  are  true  patriots,'  might  appear  at  first  sight  to 
be  in  the  second  figure  and  faulty ;  whereas  it  is  Barbara 
with  the  premisses  transposed." 

Do  you  consider  this  resolution  of  the  above  syllo- 
gism to  be  the  correct  one  } 

242.  Examine  the  validity  of  the  following  argu- 
ments : — 

(a)  Old  Parr,  healthy  as  the  wild  animals,  attained 
the  age  of  152  years;  all  men  might  be  as  healthy  as 
the  wild  animals ;  therefore,  all  men  might  attain  to 
the  age  of  152  years. 

(yS)  Most  M  is  P, 

Most  5  is  M, 
therefore,  Some  5  is  P. 

243.  Examine  the  validity  of  the  following  argu- 
ments : — 

(i)  Since  the  end  of  poetry  is  pleasure,  that 
cannot  be  unpoetical  with  which  all  are  pleased. 

(ii)  It  is  quite  absurd  to  say  "  I  would  rather  not 
exist  than  be  unhappy,"  for  he  who  says  "  I  will  this, 
rather  than  that,"  chooses  something.  Non-existence, 
however,  is  no  something,  but  nothing,  and  it  is 
impossible  to  choose  rationally  when  the  object  to  be 
chosen  is  nothing. 

244.  Can  the  following  arguments  be  reduced  to 
syllogistic  form } 


SYLLOGISMS. 


263 


CHAP.  X.] 

(i)     The  sun  is  a  thing  insensible ; 

The  Persians  worship  the  sun ; 

Therefore,  the  Persians  worship  a  thing  insensible. 

(2)  The  Divine  law  commands  us  to  honour 
kings ; 

Louis  XIV.  is  a  king  ; 

Therefore,  the  Divine  law  commands  us  to  honour 
Louis  XIV.  {Port  Royal  Logic^ 

245.  Examine  the  following  arguments;  where 
they  are  valid,  reduce  them  if  you  can  to  syllogistic 
form ;  and  where  they  are  invalid,  explain  the  nature 
of  the  fallacy : — 

(i)     We  ought  to  believe  the  Scripture ; 

Tradition  is  not  Scripture ; 

Therefore,  we  ought  not  to  believe  tradition. 

(2)  Every  good  pastor  is  ready  to  give  his  life 
for  his  sheep ; 

Now,  there  are  few  pastors  in  the  present  day  who 
are  ready  to  give  their  lives  for  their  sheep  ; 

Therefore,  there  are  in  the  present  day  few  good 
pastors. 

(3)  Those  only  who  are  friends  of  God  are  happy ; 

Now,  there  are  rich  men  who  are  not  friends  of 
God ; 

Therefore,  there  are  rich  men  who  are  not  happy. 

(4)  The  duty  of  a  Christian  is  not  to  praise  those 
who  commit  criminal  actions ; 


264 


SYLLOGISMS. 


[part  III. 


Now,  those  who  engage  in  a  duel  commit  a  criminal 
action ; 

Therefore,  it  is  the  duty  of  a  Christian  not  to 
praise  those  who  engage  in  duels. 

(5)  The  gospel  promises  salvation  to  Christians ; 
Some  wicked  men  are  Christians ; 

Therefore,  the  gospel  promises  salvation  to  wicked 
men. 

(6)  He  who  says  that  you  are  an  animal  speaks 
truly ; 

He  who  says  that  you  are  a  goose  says  that  you 
are  an  animal ; 

Therefore,  he  who  says  that  you  are  a  goose  speaks 
truly. 

(7)  You  are  not  what  I  am ; 
I  am  a  man  ; 

Therefore,  you  are  not  a  man. 

(8)  We  can  only  be  happy  in  this  world  by  aban- 
doning ourselves  to  our  passions,  or  by  combating 
them; 

If  we  abandon  ourselves  to  them,  this  is  an  un- 
happy state,  since  it  is  disgraceful,  and  we  could  never 
be  content  with  it ; 

If  we  combat  them,  this  is  also  an  unhappy  state, 
since  there  is  nothing  more  painful  than  that  inward 
war  which  we  are  continually  obliged  to  carry  on  with 
ourselves ; 

Therefore,  we  cannot  have  in  this  life  true  happi- 
ness. 


CHAP.  X.] 


SYLLOGISMS. 


265 


(9)  Either  our  soul  perishes  with  the  body,  and 
thus,  having  no  feelings,  we  shall  be  incapable  of  any 
evil ;  or  if  the  soul  survives  the  body,  it  will  be  more 
happy  than  it  was  in  the  body ; 

Therefore,  death  is  not  to  be  feared. 

[Port  Royal  Logic^ 

246.     Examine  the  following  arguments  : — 

(i)  "He  that  is  of  God  heareth  my  words :  ye 
therefore  hear  them  not,  because  ye  are  not  of  God." 

(2)  All  the  fish  that  the  net  inclosed  were  an  in- 
discriminate mixture  of  various  kinds :  those  that 
were  set  aside  and  saved  as  valuable,  were  fish  that 
the  net  enclosed  :  therefore,  those  that  were  set  aside 
and  saved  as  valuable,  were  an  indiscriminate  mixture 
of  various  kinds. 

(3)  Testimony  is  a  kind  of  evidence  which  is  very 
likely  to  be  false:  the  evidence  on  which  most  men 
believe  that  there  are  pyramids  in  Egypt  is  testimony: 
therefore,  the  evidence  on  which  most  men  believe 
that  there  are  pyramids  in  Egypt  is  very  likely  to  be 
false. 

(4)  If  Paley's  system  is  to  be  received,  one  who 
has  no  knowledge  of  a  future  state  has  no  means  of 
distinguishing  virtue  and  vice :  now  one  who  has  no 
means  of  distinguishing  virtue  and  vice  can  commit 
no  sin :  therefore,  if  Paley's  system  is  to  be  received, 
one  who  has  no  knowledge  of  a  future  state  can 
commit  no  sin. 

(5)  If  Abraham  were  justified,  it  must  have  been 
either  by  faith  or  by  works  :  now  he  was  not  justified 


^-Uk 


t^    '^^i 


nil 


•  iA;-,.. 


J-   __. 


264 


SYLLOGISMS. 


[part  III. 


Now,  those  who  engage  in  a  duel  commit  a  criminal 
action ; 

Therefore,  it  is  the  duty  of  a  Christian  not  to 
praise  those  who  engage  in  duels. 

(5)  The  gospel  promises  salvation  to  Christians ; 
Some  wicked  men  are  Christians ; 

Therefore,  the  gospel  promises  salvation  to  wicked 
men. 

(6)  He  who  says  that  you  are  an  animal  speaks 
truly ; 

He  who  says  that  you  are  a  goose  says  that  you 
are  an  animal ; 

Therefore,  he  who  says  that  you  are  a  goose  speaks 
truly. 

(7)  You  are  not  what  I  am  ; 
I  am  a  man  ; 

Therefore,  you  are  not  a  man. 

(8)  We  can  only  be  happy  in  this  world  by  aban- 
doning ourselves  to  our  passions,  or  by  combating 
them; 

If  we  abandon  ourselves  to  them,  this  is  an  un- 
happy state,  since  it  is  disgraceful,  and  we  could  never 
be  content  with  it ; 

If  we  combat  them,  this  is  also  an  unhappy  state, 
since  there  is  nothing  more  painful  than  that  inward 
war  which  we  are  continually  obliged  to  carry  on  with 
ourselves ; 

Therefore,  we  cannot  have  in  this  life  true  happi- 
ness. 


CHAP.  X.] 


SYLLOGISMS. 


265 


(9)  Either  our  soul  perishes  with  the  body,  and 
thus,  having  no  feelings,  we  shall  be  incapable  of  any 
evil ;  or  if  the  soul  survives  the  body,  it  will  be  more 
happy  than  it  was  in  the  body ; 

Therefore,  death  is  not  to  be  feared. 

[Port  Royal  Lo^ic] 

246.     Examine  the  following  arguments  : — 
(i)     "He  that  is  of  God  heareth  my  words :  ye 
therefore  hear  them  not,  because  ye  are  not  of  God." 

(2)  All  the  fish  that  the  net  inclosed  were  an  in- 
discriminate mixture  of  various  kinds:  those  that 
were  set  aside  and  saved  as  valuable,  were  fish  that 
the  net  enclosed  :  therefore,  those  that  were  set  aside 
and  saved  as  valuable,  were  an  indiscriminate  mixture 
of  various  kinds. 

(3)  Testimony  is  a  kind  of  evidence  which  is  very 
likely  to  be  false:  the  evidence  on  which  most  men 
believe  that  there  are  pyramids  in  Egypt  is  testimony: 
therefore,  the  evidence  on  which  most  men  believe 
that  there  are  pyramids  in  Egypt  is  very  likely  to  be 
false. 

(4)  If  Faley's  system  is  to  be  received,  one  who 
has  no  knowledge  of  a  future  state  has  no  means  of 
distinguishing  virtue  and  vice :  now  one  who  has  no 
means  of  distinguishing  virtue  and  vice  can  commit 
no  sin :  therefore,  if  Paley's  system  is  to  be  received, 
one  who  has  no  knowledge  of  a  future  state  can 
commit  no  sin. 

(5)  If  Abraham  were  justified,  it  must  have  been 
either  by  faith  or  by  works  :  now  he  was  not  justified 


266 


SYLLOGISMS. 


[part  III. 


by  faith  (according  to  James),  nor  by  works  (accord- 
ing to  Paul):  therefore,  Abraham  was  not  justified. 

(6)  For  those  who  are  bent  on  cultivating  their 
minds  by  diligent  study,  the  incitement  of  academical 
honours  is  unnecessary  ;  and  it  is  ineffectual,  for  the 
idle,  and  such  as  are  indifferent  to  mental  improve- 
ment :  therefore,  the  incitement  of  academical  honours 
is  either  unnecessary  or  ineffectual. 

(7)  He  who  is  most  hungry  eats  most ;  he  who 
eats  least  is  most  hungry :  therefore,  he  who  eats  least 
eats  most. 

(8)  A  monopoly  of  the  sugar-refining  business  is 
beneficial  to  sugar- refiners  :  and  of  the  corn-trade  to 
corn-growers:  and  of  the  silk-manufacture  to  silk- 
weavers,  &c.,  &c. ;  and  thus  each  class  of  men  are 
benefited  by  some  restrictions.  Now  all  these  classes 
of  men  make  up  the  whole  community :  therefore  a 
system  of  restrictions  is  beneficial  to  the  community. 

[Whately.] 

247.  The  following  are  a  few  examples  in  which 
the  reader  can  try  his  skill  in  detecting  fallacies, 
determining  the  peculiar  form  of  syllogisms,  and  sup- 
plying the  suppressed  premisses  of  enthymemes. 
Several  of  the  examples  contain  more  than  one 
syllogism. 

(i)  None  but  those  who  are  contented  with  their 
lot  in  life  can  justly  be  considered  happy.  But  the 
truly  wise  man  will  always  make  himself  contented 
with  his  lot  in  life,  and  therefore  he  may  justly  be 
considered  happy. 


CHAP.  X.] 


SYLLOGISMS. 


267 


(2)  All  intelligible  propositions  must  be  either 
true  or  false.  The  two  propositions  "Caesar  is  living 
still,"  and  "Caesar  is  dead,"  are  both  intelligible  pro- 
positions ;  therefore  they  are  both  true,  or  both  false. 

(3)  Many  things  are  more  difficult  than  to  do 
nothing.  Nothing  is  more  difificult  to  do  than  to 
walk  on  one's  head.  Therefore,  many  things  are 
more  difificult  than  to  walk  on  one's  head. 

(4)  None  but  Whigs  vote  for  Mr  B.  All  who 
vote  for  Mr  B.  are  ten-pound  householders.  There- 
fore none  but  Whigs  are  ten-pound  householders. 

(5)  If  the  Mosaic  account  of  the  cosmogony  is 
strictly  correct,  the  sun  was  not  created  till  the 
fourth  day.  And  if  the  sun  was  not  created  till  the 
fourth  day,  it  could  not  have  been  the  cause  of  the 
alternation  of  day  and  night  for  the  first  three  days. 
But  either  the  word  *'  day  "  is  used  in  Scripture  in  a 
different  sense  to  that  in  which  it  is  commonly  ac- 
cepted now,  or  else  the  sun  must  have  been  the  cause 
of  the  alternation  of  day  and  night  for  the  first  three 
days.  Hence  it  follows  that  either  the  Mosaic  account 
of  the  cosmogony  is  not  strictly  correct,  or  else  the 
word  "day"  is  used  in  Scripture  in  a  different  sense 
to  that  in  which  it  is  commonly  accepted  now. 

(6)  Suffering  is  a  title  to  an  excellent  inheritance; 
for  God  chastens  every  son  whom  He  receives. 

(7)  It  will  certainly  rain,  for  the  sky  looks  very 
black.  [Solly,  Syllabus  of  Logic] 


268  SYLLOGISMS.  [part  hi 

248.  Examine  the  following  arguments: 

(i)  All  the  householders  in  the  kingdom,  except 
women,  are  legally  electors,  and  all  the  male  house- 
holders are  precisely  those  men  who  pay  poor-rates ; 
it  follows  that  all  men  who  pay  poor-rates  are  electors. 

(2)  All  men  are  mortals,  and  all  mortals  are 
those  who  are  sure  to  die ;  therefore,  all  men  are 
all  those  who  are  sure  to  die. 

[Jevons,  Studies,  p.  162.] 

249.  State  the  following  arguments  in  Logical 
form,  and  examine  their  validity  : — 

(i)  Poetry  must  be  either  true  or  false:  if  the 
latter,  it  is  misleading ;  if  the  former,  it  is  disguised 
history,  and  savours  of  imposture  as  trying  to  pass 
itself  off  for  more  than  it  is.  Some  philosophers 
have  therefore  wisely  excluded  poetry  from  the  ideal 
commonwealth. 

(2)  If  we  never  find  skins  except  as  the  tegu- 
ments of  animals,  we  may  safely  conclude  that 
animals  cannot  exist  without  skins.  If  colour  can- 
not exist  by  itself,  it  follows  that  neither  can  any- 
thing that  is  coloured  exist  without  colour.  So  if 
language  without  thought  is  unreal,  thought  without 
language  must  also  be  so. 

(3)  Had  an  armistice  been  beneficial  to  France 
and  Germany,  it  would  have  been  agreed  upon  by 
those  powers;  but  such  has  not  been  the  case;  it  is 
plain  therefore  that  an  armistice  would  not  have  been 
advantageous  to  either  of  the  belligerents. 


CHAP.  X.] 


SYLLOGLSMS. 


269 


(4)    If  we  are  marked  to  die,  we  are  enow 
To  do  our  country  loss :  and,  if  to  live, 
The  fewer  men,  the  greater  share  of  honour. 

[o.] 

250.  Dr  Johnson  remarked  that  "a  man  who 
sold  a  penknife  was  not  necessarily  an  ironmonger." 
Against  what  logical  fallacy  was  this  remark  directed.^ 

[c] 

251.  Exhibit  the  following  in  syllogistic  form; 
naming  the  mood  and  figure ;  when  possible,  reduce 
them  to  the  first  figure :  (a)  The  disciples  of  Wagner 
overrate  him,  for  he  has  caused  a  great  reform  in 
dramatic  art,  and  all  great  reformers  are  over-esti- 
mated by  their  followers,  (b)  Some  undergraduates 
are  guilty  of  conduct  to  which  no  gentleman  would 
stoop ;  so  some  undergraduates  are  not  gentlemen. 
[c)  Not  all  the  things  we  neglect  are  worthless,  for 
some  truths  are  neglected  and  none  without  value. 

[c] 

252.  Examine  on  logical  principles  the  following 
arguments ;  and,  if  you  find  any  fallacies,  name  them : 

(a)  The  existence  of  State-officials  is  unjustifi- 
able: for  since  men  are  by  nature  equal,  it  is  con- 
trary to  nature  that  one  should  govern  another. 

ip)  Instinct  and  reason  are  opposed  :  so  a  good 
action,  if  instinctive,  is  the  opposite  of  that  which 
reason  would  dictate.  [c] 

253.  Put  the  following  propositions  into  their 
simplest  Logical  form;  name  the  Syllogistic  Moods 


270  SYLLOGISMS.  [part  hi. 

in  which  they  can  be  proved ;  and  find  premisses  that 
in  some  Mood  will  prove  them  : 

(i)     Not  all  the  unhappy  are  evildoers. 

(2)     Only  the  wise  are  free.  [c] 

254.  Examine  the  following  arguments,  pointing 
out  any  fallacies  that  they  contain : 

(a)  The  more  correct  the  logic,  the  more  certainly 
will  the  conclusion  be  wrong  if  the  premisses  are 
false.  Therefore,  where  the  premisses  are  wholly  un- 
certain the  best  logician  is  the  least  safe  guide. 

(d)  The  spread  of  education  among  the  lower 
orders  will  make  them  unfit  for  their  work :  for  it  has 
always  had  that  effect  on  those  among  them  who 
happen  to  have  acquired  it  in  previous  times. 

(c)  This  pamphlet  contains  seditious  doctrines. 
The  spread  of  seditious  doctrines  may  be  dangerous 
to  the  State.  Therefore,  this  pamphlet  must  be  sup- 
pressed. [C.] 

255.  "  To  prove  that  Dissent  is  wrong  you  must 
appeal  to  the  authority  of  the  Church,  and  this  you 
must  base  on  the  Bible ;  and  you  must  also  deny  the 
supremacy  of  Conscience.  Moreover  you,  at  least,  as 
an  Anglican,  must  ignore  the  Reformation." 

How  should  you  draw  out  fully  the  argument  here 
implied.^  To  what  extent  does  it  naturally  fall  into 
syllogistic  form  ?  [yj 

256.  No  one  can  maintain  that  all  republics 
secure  good   government   who   bears    in    mind    that 


CHAP.  X.]  SYLLOGISMS.  271 

good  government   is   inconsistent   with   a   licentious 
press. 

What  premisses  must  be  supplied  to  express  the 
above  reasoning  in  Ferio^  Festino  and  Ferison  re- 
spectively 1  [v.] 

257.  Using  any  of  the  forms  of  Immediate 
Inference,  shew  in  how  many  moods  the  following 
argument  can  be  expressed  : — "  Every  law  is  not 
binding,  for  some  laws  are  morally  bad,  and  nothing 
which  is  so  is  binding."  [l.] 

258.  State  the  following  reasonings  in  strict 
logical  form,  and  estimate  their  validity  : — 

(a)  As  thought  is  existence,  what  contains  no 
element  of  thought  must  be  non-existent. 

(h)  Since  the  laws  allow  everything  that  is  inno- 
cent, and  avarice  is  allowed,  it  is  innocent. 

(c)  Timon  being  miserable  is  an  evil-doer,  as 
happiness  springs  from  well-doing.  [l.] 

259.  Comment  carefully  upon  the  following  state- 
ments : — 

"  The  most  perfect  Logic  will  not  serve  a  man 
who  starts  from  a  false  premiss." 

"  I  am  enough  of  a  logician  to  know  that  from 
false  premisses  it  is  impossible  to  draw  a  true  conclu- 


sion. 


)) 


[L.] 


260.  Might  I  be  satisfied  that  a  particular  war 
was  a  just  one,  assuming  (what  was  the  fact)  that  it 
was  popular,  and  also  (what  is  more  doubtful)  that  all 
just  wars  are  popular } 


2/2 


SYLLOGISMS. 


[part  111. 


Are  honours  and  rewards,  public  or  private,  to  be 
pronounced  useless,  because  they  cannot  influence  the 
stupid,  and  men  of  genius  rise  above  them  ? 

Because  some  persons  in  the  dark  cannot  help 
thinking  of  ghosts,  though  they  do  not  believe  in 
them,  does  it  follow  that  it  is  absurd  to  maintain  that, 
when  we  cannot  avoid  thinking  or  conceiving  of  a 


thing,  it  must  be  true  ? 


[L.] 


CHAPTER  XL 


PROBLEMS   ON    THE   SYLLOGISM. 


261.  Prove  by  means  of  the  syllogistic  rules 
that,  given  the  truth  of  one  premiss  and  of  the 
conclusion  of  a  valid  syllogism,  the  knowledge  thus 
in  our  possession  is  in  no  case  sufficient  to  prove  the 
truth  of  the  other  premiss. 

We  have  to  shew  that  if  one  premiss  and  the  conclusion 
of  a  valid  syllogism  be  taken  as  a  new  pair  of  premisses 
they  do  not  in  any  case  suffice  to  establish  the  other 
premiss. 

T/ie  premiss  given  true  must  be  affirmative^  for  if  it  is 
negative,  the  original  conclusion  will  be  negative,  and  com- 
bining these  we  shall  have  two  negative  premisses  which 
can  yield  no  conclusion. 

The  middle  term  7nust  be  dist?'ibiited  in  the  premiss  given 
true,  for  if  not  it  must  be  distributed  in  the  other  premiss, 
but  this  being  the  conclusion  of  the  new  syllogism,  it  must 
also  be  distributed  in  the  premiss  given  true  or  we  shall 
have  an  illicit  process  in  the  new  syllogism. 

Therefore,  the  premiss  given  true,  being  affirmative,  and 
distributing  the  middle  term,  cannot  distribute  the  other 
term  which  it  contains.  Neither  therefore  can  this  term 
be  distributed  in  the  original  conclusion.     But  this  is  the 


K.  L. 


i8 


274 


SYLLOGISMS. 


[part  III. 


term  which  will  be  the  middle  term  of  the  new  syllogism, 
and  7ae  shall  therefore  have  undistributed  middle. 

The  given  syllogism  then  being  valid,  we  have  shewn 
it  to  be  impossible  that  a  new  syllogism  having  one  of  the 
original  premisses  and  the  original  conclusion  for  its  pre- 
misses, with  the  other  original  premiss  for  its  conclusion, 
can  be  valid  also\ 

262.  Given  that  in  a  valid  syllogism  one  premiss 
is  false  and  the  other  true,  shew  that  in  no  case  will 
this  suffice  to  prove  the  conclusion  false^ 

This  might  be  established  by  taking  all  possible  syllo- 
gisms, and  shewing  that  the  statement  holds  true  with 
regard  to  each  in  turn  ;  but  this  method  is  clearly  to  be 
avoided  if  possible. 

It  might  also  be  deduced  from  the  proposition  established 
in  the  preceding  example.  Let  the  premisses  of  a  valid 
syllogism  be  P  and  Q  and  the  conclusion  R.  P  and  the 
contradictory  of  Q  will  not  prove  the  contradictory  of  R ; 
for  if  so  it  would  follow  that  P  and  R  would  prove  Q ;  but 
this  has  been  shewn  not  to  be  the  case. 

Another  easy  solution  is  obtainable  by  assuming  that 

^  Other  methods  of  solution  more  or  less  distinct  from  the  above 
might  be  given.  A  somewhat  similar  problem  is  discussed  by  Solly, 
Syllabus  of  Logic,  pp.  123 — 126,  132 — 136.  Hamilton  {LogiCy  i.  p.  450) 
considers  the  doctrine  "that  if  the  conclusion  of  a  syllogism  be  tnie,  the 
premisses  may  be  either  true  or  false,  but  that  if  the  conclusion  be  false, 
one  or  both  of  the  premisses  must  be  false"  to  be  extra-logical,  if 
it  is  not  absolutely  erroneous.  He  is  clearly  wrong,  since  the  doctrine 
in  question  admits  of  a  purely  formal  proof. 

■^  This  problem  might  also  be  stated  as  follows, — Shew  that  if  for 
one  of  the  premisses  of  a  valid  syllogism  we  substitute  its  contradictory, 
this  will  not  in  any  case  enable  us  to  establish  the  contradictory  of  the 
original  conclusion. 

6    i^^X^H 


T 


y 


j-^ 


CHAP.  XI.] 


SYLLOGISMS. 


275 


the  given  syllogism  is  reduced  to  Figure  i.  After  such  re- 
duction, it  will,  in  accordance  with  the  special  rules  of 
Figure  i,  have  a  universal  major  and  an  affirmative  minor. 
Then  since  the  contradictory  of  a  universal  is  particular  and 
of  an  affirmative  negative,  if  either  premiss  is  given  false 
we  have  in  its  place  either  a  particular  major  or  a  negative 
minor.  But,  (since  the  syllogism  is  still  in  Figure  i),  in 
neither  of  these  cases  can  we  draw  any  conclusion  at  all, 
and  therefore  a  fortiori  we  cannot  infer  that  the  original 
conclusion  is  false. 

I  add  an  outline  of  an  independent  general  solution  of 
the  given  problem  \ 

Let  the  following  symbols  be  used: — 

T  -  premiss  given  true; 
F  =  premiss  given  false; 
C  =  original  conclusion; 
F'  =  contradictory  of  F\ 
C  =  contradictory  of  C; 
a  ^  original  syllogism; 

)8  =  syllogism  of  which  the  premisses  are  T  and  F\  and 
the  conclusion  C"; 
P  =  major  term; 
M  =  middle  term ; 
aS*  ^  minor  term. 

We  have  to  shew  that  /?  cannot  be  a  valid  syllogism. 

T  cannot  be  particular,  for  in  this  case  F'  would  also 
be  particular. 

T  cannot  be  negative,  for  in  this  case  F'  would  also  be 
negative. 

T  then  must  be  universal  affirmative. 

^  Several  steps  are  omitted,  but  these  the  student  should  carefully 
fill  in  for  himself. 

18—2 


These  will  be  the  same  both  in 
a  and  ^. 


276 


SYLLOGISMS. 


[PART  III. 


(i)     Let  F  also  be  universal  affirmative. 

We  may  shew  that  C  must  also  be  universal,  (/'.  ^.,  a  can- 
not have  a  weakened  conclusion) ;  and  it  must  of  course  be 
affirmative. 

Then  in  a,  ^  and  M  must  be  distributed; 
in  ^,  P  and  M  must  be  distributed. 

But  if  F  distributed  M^  M  cannot  be  distributed  in  /? ; 
and  if /^ distributed  6",  /^cannot  be  distributed  in  p. 

(2)  Let  F  be  universal  negative.  We  may  again  shew 
that  C  must  be  universal. 

In  this  case  T  cannot  distribute  M)  but  neither  can  F' 
distribute  M. 

(3)  Let  F  be  particular  affirmative. 

C  will  be  universal  negative.  Therefore,  in  /3  we  must 
distribute  S,  M,  P. 

But  T  must  distribute  M ;  it  cannot  therefore  distribute 
6*  or  /*,  one  of  which  must  therefore  be  undistributed  in  ft, 

(4)  Let  F  be  particular  negative. 
In  a,  M  and  P  must  be  distributed ; 
in  j8,  M  and  S  must  be  distributed. 

But  if  F  distributed  M^  M  cannot  be  distributed  in  ^ ; 
and  if  F  distributed  P^  S  cannot  be  distributed  in  fi. 

263.  Given  a  valid  syllogism  in  Figure  i,  is  there 
any  case  in  w^hich  the  mere  knowledge  that  we  may 
start  from  the  contradiction  of  its  premisses  will 
furnish  premisses  for  another  valid  syllogism  } 

264.  An  apparent  syllogism  of  the  second  figure 
with  a  particular  premiss  is  found  to  break  the  general 
rules  of  the  syllogism  in  this  particular  only,  that  the 
middle  term  is  undistributed.     If  the  particular  prc- 


CHAP.  XI.]  SYLLOGISMS.  277 

miss  is  false  and  the  other  true,  what  do  we  know 
about  the  truth  or  falsity  of  the  conclusion  } 

Can  an  apparent  syllogism  break  all  the  rules  of 
syllogism  at  once  ? 

265.  Given  the  two  following  statements  false: — 
(i)  either  all  M  is  all  P,  or  some  M  is  not  P\ 
(ii)  some  5  is  not  M\ — what  is  all  that  you  can 
infer,  (a)  with  regard  to  S  in  terms  of  P\  {b)  with 
regard  to  P  in  terms  of  vS  ? 

266.  If  (i)  it  is  false  that  whenever  X  is  found 
Y  is  found  with  it,  and  (2)  not  less  untrue  that  X  is 

sometimes  found  without  the  accompaniment  of  Z^ 
are  you  justified  in  denying  that  (3)  whenever  Z  is 
found  there  also  you  may  be  sure  of  finding  Yt  And 
however  this  may  be,  can  you  in  the  same  circum- 
stances judge  anything  about  Y  in  terms  of  Z }    [r.] 

267.  If  whenever  X  is  present,  Z  is  not  absent, 
and  sometimes  when  Y  is  absent,  X  is  present,  but  if 
it  cannot  be  said  that  the  absence  of  X  determines 
anything  about  either  Y  or  Z^  can  anything  be  deter- 
mined as  between  Z  and  Yt  [r.] 

268.  If  ^  is  always  found  to  coexist  with  A^ 
except  when  X  is  Y,  (which  it  commonly,  though  not 
always,  is),  and  if,  even  in  the  few  cases  where  X  is 
not  F,  C  is  never  found  absent  without  B  being 
absent  also,  can  you  make  any  other  assertion  about 

C  ?  [R.] 


278 


SYLLOGISMS. 


[part  III. 


269.  From  P  follows  Q ;  and  from  R  follows  5 ; 
but  Q  and  6"  cannot  both  be  true ;  shew  that  P  and 
R  cannot  both  be  true.  (De  Morgan.) 

270.  Given  a  syllogism,  shew  in  what  cases  it  is 
possible  to  reach  the  same  conclusion  by  substituting 
for  the  middle  term  its  contradictory.  [w.] 

[We  are  supposed  here  to  perform  immediate  inferences 
upon  our  premisses  so  as  to  obtain  a  new  middle  term 
which  is  the  contradictory  of  the  original  middle  term.] 

271.  What  conclusion  can  be  drawn  from  the 
following  propositions  ? 

The  members  of  the  board  were  all  either  bond- 
holders or  shareholders,  but  not  both ;  and  the  bond- 
holders, as  it  happened,  were  all  on  the  board,    [v.] 

We  have  given, — 

No  member  of  the  board  is  both  a  bondholder  and 
a  shareholder, 

All  bondholders  are  members  of  the  board; 
and  these  premisses  yield  a  conclusion  (in  Celarent), 

No  bondholder  is  both  a  bondholder  and  a  shareholder, 

that  is.  No  bondholder  is  a  shareholder. 

272.  The  following  rules  were  drawn  up  for  a 
club  : — 

(i)  The  financial  committee  shall  be  chosen  from 
amongst  the  general  committee ; 

(ii)  No  one  shall  be  a  member  both  of  the  general 
and  library  committees,  unless  he  be  also  on  the 
financial  committee ; 


CHAP.  XI.] 


SYLLOGISMS. 


279 


(iii)     No  member  of  the  library  committee  shall 

be  on  the  financial  committee. 

Is  there  anything  self-contradictory  or  superfluous 

in  these  rules  ^ 

[Venn,  Symbolic  Logic,  pp.  261 — 264.] 

Let  F  =-  member  of  the  financial  committee, 
G  =  member  of  the  general  committee, 
Z  =  member  of  the  library  committee. 

The  above  rules  then  become, — 
(i)     All  Pis  G; 
(ii)     If  Z  is  G,  itis  P; 
(iii)     No  Z  is  P. 
From  (ii)  and  (iii)  we  obtain 

(iv)     No  Z  is  C. 
The  rules  may  therefore  be  written, 
(i)     All  P  is  G, 

(2)  No  Z  is  G, 

(3)  No  Z  is  P 

But  in  this  form  (3)  is  deducible  from  (i)  and  (2). 

All  that  is  contained  therefore  in  the  rules  as  originally 
stated  may  be  expressed  by  (i)  and  (2);  that  is,  the  rules 
as  originally  stated  were  partly  superfluous,  and  they  may 
be  reduced  to 

(i)  The  financial  committee  shall  be  chosen  from 
amongst  the  general  committee ; 

(2)  No  one  shall  be  a  member  both  of  the  general 
and  library  committees. 

If  (ii)  is  interpreted  as  implying  that  there  are  individuals 
who  are  on  both  the  general  and  library  committees,  then 
it  follows  that  (ii)  and  (iii)  are  inconsistent  with  each 
other. 


28o 


SYLLOGISMS. 


[part  III. 


273.  Are  assumptions  with  regard  to  "existence" 
involved  in  any  of  the  syllogistic  processes  ? 

We  may  as  in  section  104  take  three  distinct  suppositions 
with  regard  to  the  existential  implication  of  propositions, 
and  proceed  to  answer  the  above  question  on  the  basis  of 
each  in  turn.     The  three  suppositions  are: — 

(i)  All  propositions  imply  the  existence  both  of  their 
subjects,  and  of  their  predicates. 

(2)  No  propositions  imply  the  existence  either  of  their 
subjects  or  of  their  predicates. 

(3)  Particular  propositions  imply  the  existence  of  their 
subjects ;  but  universal  propositions  do  not. 

J^irsf,  we  may  take  the  supposition  that  every  proposition 
implies  the  existence  both  of  its  subject  and  of  its  predicate.  In 
this  case,  the  existence  of  the  major,  middle  and  minor 
terms  is  guaranteed  by  the  premisses,  and  therefore  no 
further  assumption  with  regard  to  existence  is  required  in 
order  that  the  conclusion  may  be  legitimately  obtained  \ 

Secondly^  we  may  take  the  supposition  that  no  proposition 
logically  implies  the  existence  either  of  its  subject  or  of  its  predi- 
cate. Let  the  major,  middle  and  minor  terms  be  respectively 
P,  J/,  S.  The  conclusion  will  imply  that  if  there  is  any  S 
there  is  some  P  or  not-y,  (according  as  it  is  affirmative  or 
negative).     Will  the  premisses  also  necessarily  imply  this  ? 

It  has  been  shewn  in  section  141  that  a  universal 
affirmative  conclusion,  All  5  is  P,  can  only  be  proved  by 
means  of  the  premisses, — All  M  is  P,  All  ^S"  is  M  \  and  it  is 
clear  that  these  premisses  themselves  necessarily  imply  that 

^  If  however  we  are  to  be  allowed  to  proceed  as  in  section  123, 
(where  from  all  P  is  M,  all  6"  is  M^  we  inferred  that  some  not-^  is 
not-/'),  we  must  posit  the  existence  not  merely  of  the  terms  directly 
involved,  but  also  of  their  contradictories. 


CHAP.  XI.] 


SYLLOGISMS. 


281 


if  there  is  any  S  there  is  some  P.  No  assumption  then 
with  regard  to  existence  is  involved  in  syllogistic  reasoning 
if  the  conclusion  is  universal  affirmative. 

Again,  as  shewn  in  section  141,  a  universal  negative 
conclusion,  No  S  is  /*,  can  only  be  proved  in  the  following 
w^ays, — 

(i)     No  M  is  P,  (or  No  P  is  M\ 
All  S  is  J/, 

therefore.  No  S  is  P. 

(ii)     All  P  is  J/, 

No  6"  is  M,  (or  No  M\^  S\ 

therefore,  No  S  is  P. 

In  (i)  the  minor  premiss  implies  that  if  S  exists  then  M 
exists,  and  the  major  premiss  that  if  J/exists  then  not-^ exists. 

In  (ii)  the  minor  premiss  implies  that  if  6"  exists  then 
not- J/  exists,  and  the  major  premiss  that  if  not- J/  exists 
then  not-/' exists,  (as  shewn  in  section  104). 

It  follows  then  that  no  assumption  is  involved  if  the 
conclusion  is  universal  negative. 

Next,  let  the  conclusion  be  particular.  The  implication 
of  the  conclusion  with  regard  to  existence  is  now  contained 
in  the  premisses  themselves,  if  the  minor  premiss  is  affirma- 
tive, and  if  the  minor  term  is  the  subject  of  the  minor 
premiss,  and  the  middle  term  the  subject  of  the  major 
prjmiss,  (i.c.^  if  the  syllogism  is  in  Figure  i).  The  same  will 
be  found  to  hold  good  on  special  examination  of  the  moods 
of  Figure  2  which  yield  particular  conclusions.  But  it  is 
otherwise  with  regard  to  the  moods  of  Figures  3  and  4. 
Take,  for  example,  a  syllogism  in  Darapti^ — 

All  M  is  P, 
AllJ/is5, 

therefore.  Some  6"  is  P. 


282 


SYLLOGISMS. 


[part  hi. 


The  conclusion  implies  that  if  S  exists  P  exists;  but 
consistently  with  the  premisses,  S  may  be  existent  while 
^ and  Pare  both  non-existent.  An  implication  is  therefore 
contained  in  the  conclusion  which  is  not  contained  in  the 
premisses  themselves. 

Our  results  may  now  be  summed  up  as  follows: — On 
the  supposition  that  no  proposition  logically  implies  the 
existence  either  of  its  subject  or  of  its  predicate,  we  do  not 
require  to  7tiake  any  assumption  loith  regard  to  existence  in 
any  syllogistic  process  yielding  a  universal  conclusion  in  what- 
ever figure  it  may  be,  nor  i?i  any  syllogistic  process  yielding 
a  particular  conclusion  provided  it  is  in  Figure  i  or  Figure  2  ; 
but  it  is  otherwise  if  a  particular  conclusion  is  obtained  in 
Figure  3  or  Figure  4. 

Thirdly^  taking  the  supposition  that  particular  proposi- 
tions imply  the  existence  of  their  subjects,  although  universal 
propositions  do  not,  it  will  be  found  that  assumptions  with 
regard  to  existence  are  involved  in  syllogistic  reasoning  in 
the  following  and  only  in  the  following  cases, — 

(i)     In  Figures  2  and  4,  if  the  conclusion  is  particular  ; 

(ii)  In  Figures  i  and  3,  if  the  minor  premiss  is  universal 
and  the  conclusion  particular. 

The  student  should  for  himself  fill  in  the  steps  necessary 
to  establish  this  conclusion. 

274.  "■  Whatever  P  and  Q  may  stand  for,  we  may 
shew  a  priori  that  some  P  is  Q.  For  All  PQ  is  Q  by 
the  law  of  identity,  and  similarly  All  PQ  is  P\ 
therefore,  by  a  syllogism  in  Dm-apti,  some  P  is  Qr 
How  would  you  deal  with  this  paradox } 

A  solution  is  afforded  by  the  discussion  contained  in 
the  preceding  section;  and  this  example  seems  to  shew 
that  the  enquiry, — how  far  assumptions  with  regard  to  exist- 


CHAP.  XL]  SYLLOGISMS.  283 

ence  are  involved  in  syllogistic  processes, — is  not  irrelevant 
or  unnecessary. 

275.  If  P  is  0,  and  Q  is  R,  it  follows  that  P  is 
R ;  but  suppose  it  to  be  discovered  that  no  such  thing 
as  Q  exists, — How  is  the  truth  of  the  conclusion,  P 
is  R,  afTfected  by  this  discovery  }  [l.] 

276.  De  Morgan  says : — "  In  all  syllogisms  the 
existence  of  the  middle  term  is  a  datimi"  Inquire 
into  the  accuracy  of  this  assertion.  What  does 
existence  here  mean  }  [l.] 

277.  On  the  supposition  that  no  proposition 
logically  implies  the  existence  either  of  its  subject  or 
of  its  predicate,  find  in  what  cases  of  the  Reduction 
of  Syllogisms  to  Figure  i  assumptions  with  regard 
to  existence  are  involved. 

278.  Given  that  the  middle  term  is  distributed 
twice  in  the  premisses  of  a  syllogism,  determine 
directly,  {i.e.,  without  any  reference  to  the  special 
rules  of  the  figures,  or  the  possible  moods  in  each 
figure),  in  what  different  moods  it  might  possibly  be. 

The  premisses  must  be  either  both  affirmative,  or  one 
affirmative  and  one  negative. 

In  the  first  case,  both  premisses  being  affirmative  can  dis- 
tribute their  subjects  only.  I'he  middle  term  must  therefore 
be  the  subject  in  each,  and  both  must  be  universal.  This 
limits  us  to  the  one  syllogism, — 

All  J/ is  jP; 
All  M  is  S, 

therefore.   Some  S  is  P, 


284 


SYLLOGISMS. 


[part  III. 


In  the  second  case,  one  premiss  being  negative,  the  con- 
clusion must  be  negative  and  will  therefore  distribute  the 
major  term.  Hence,  the  major  premiss  must  distribute  the 
major  term,  and  also  (by  hypothesis)  the  middle  term. 
This  condition  can  be  fulfilled  only  by  its  being  one  or 
other  of  the  following,— No  M  is  F,  or  ^o  F'lsM.  The 
major  being  negative,  the  minor  must  be  affirmative,  and  in 
order  to  distribute  the  middle  term  it  must  be  All  Mh  S, 

In  this  case  then  we  get  two  syllogisms,  namely, — 

No  J/ is  y^, 
All  M  is  S, 

therefore.   Some  S  is  not  P. 

No /^  is  J/, 
All  M  is  S, 

therefore,   Some  S  is  not  P. 

The  given  condition  limits  us  therefore  to  three  syllo- 
gisms, (one  affirmative  and  two  negative);  and  by  reference 
to  the  mnemonic  verses  we  may  now  identify  these  with 
Darapti  and  Felapton  in  Figure  3,  and  Fesapo  in  Figure  4. 

279.  If  the  major  premiss  is  affirmative,  and  if 
the  major  term  is  distributed  both  in  premisses  and 
conclusion,  while  the  minor  term  is  undistributed  in 
both,  determine  directly  the  mood  and  figure.      [n.] 

280.  If  the  major  term  be  distributed  in  the 
premisses  and  undistributed  in  the  conclusion,  deter- 
mine directly  the  mood  and  figure.  [c] 

[Professor  Jevons  gives  this  question  in  the  form:  ^*  If  the 
major  term  be  universal  in  the  premisses  and  particular  in 
the  conclusion,  determine  the  mood  and  figure,  it  being 
understood  that  the  conclusion  is  not  a  weakened  one" 


CHAP.  XI.] 


SYLLOGISMS. 


285 


{Studies  in  Deductive  Logic,  p.  103) ;  but  the  condition  here 
introduced  seems  unnecessary,  since  we  are  in  any  case 
limited  to  a  single  syllogism.] 

281.  Given  a  valid  syllogism  with  two  universal 
premisses  and  a  particular  conclusion,  such  that  if  its 
subaltern  is  substituted  for  either  of  the  premisses 
the  same  conclusion  cannot  be  inferred,  determine 
the  mood  and  figure  of  the  syllogism. 

If  there  is  such  syllogism,  let  S,  M,  F  be  its  minor, 
middle  and  major  terms  respectively. 

Since  the  conclusion  is  given  particular  it  must  be  either 
Some  5  is  F,  or  Some  S  is  not  P. 

First,  if  possible,  let  it  be  Some  S  is  P. 

The  only  term  which  we  require  to  distribute  in  the 
premisses  is  M.  But  since  we  have  two  universal  premisses, 
two  terms  must  be  distributed  in  them  as  subjects  \  One  of 
these  must  be  superfluous ;  and  therefore  for  one  of  the 
premisses  we  may  substitute  its  subaltern,  and  still  get  the 
same  conclusion. 

The  conclusion  cannot  then  be  Some  S  is  P. 

Secondly,  if  possible,  let  the  conclusion  be  Some  S  is  not 

P, 

If  the  subject  of  the  minor  premiss  is  S,  we  may  clearlv 
substitute  its  subaltern  without  affecting  the  conclusion. 
The  subject  of  the  minor  premiss  must  therefore  be  M, 
which  will  thus  be  distributed  in  this  premiss.  M  cannot 
also  be  distributed  in  the  major,  or  else  it  is  clear  that  its 
subaltern  might  be  substituted  for  the  minor  and  nevertlie- 

1  We  here  include  the  case  in  which  the  middle  term  is  itself  twice 
distributed. 


286 


SYLLOGISMS. 


[part  III. 


less  the  same  conclusion  inferred.  The  major  premiss  must 
therefore  be  afifirmative  with  M  for  its  predicate.  This 
limits  us  to  the  syllogism, — 

All  P  is  M, 
No  J/ is  6*, 

therefore,    Some  S  is  not  P\ 

and  this  syllogism,  which  is  AEO  in  Figure  4,  does  fulfil 
the  given  conditions,  for  if  either  premiss  is  made  particular, 
it  becomes  invalid. 

The  above  amounts  to  a  general  proof  of  the  proposition 
laid  down  in  section  147.  Every  syllogism  in  luhich  there 
are  two  tiniversal  premisses  with  a  partiadar  condusiofi  is  a 
strengthened  syllogism,  with  the  one  exception  0/  AKO  in 
Figure  4. 

[In  his  studies  in  Dedudive  LogiCy  p.  105,  Jevons  gives 
the  following:  *' Prove  that  wherever  there  is  a  particular 
conclusion  without  a  particular  premiss,  something  super- 
fluous is  invariably  assumed  in  the  premisses."  The  case  of 
AEO  in  Figure  4,  however,  shews  that  this  needs  qualifica- 
tion.] 

282.  Given  two  valid  syllogisms  in  the  same 
figure  in  which  the  major,  middle  and  minor  terms 
are  respectively  the  same,  shew,  without  reference  to 
the  mnemonic  verses,  that  if  the  minor  premisses  are 
subcontraries,  the  conclusions  will  be  identical. 

The  minor  premiss  of  one  of  the  syllogisms  must  be  O, 
and  the  major  premiss  of  this  syllogism  must  therefore  be 
A  and  the  conclusion  O.  The  middle  and  the  major  terms 
having  then  to  be  distributed  in  the  premisses,  this  syllogism 
is  determined,  namely, — 


CHAP.  XI.] 


SYLLOGISMS. 


287 


All  P  is  M, 
Some  6"  is  not  M, 


therefore,  Some  6*  is  not  P. 

Since  the  other  syllogism  is  to  be  in  the  same  figure,  its 
minor  premiss  must  be  Some  -S  is  J/ the  major  must  there- 
fore be  universal,  and  in  order  to  distribute  the  middle  term 
it  must  be  negative.    The  syllogism  then  is  also  determined, 

namely, — 

No  P  is  J/, 

Some  iS  is  J/, 


therefore,  Some  .S"  is  not  P. 

The  conclusions  of  the  two  syllogisms  are  thus  shewn  to 
be  identical 

283.  Given  two  valid  syllogisms  in  the  same 
figure  in  which  the  major,  middle  and  minor  terms 
are  respectively  the  same,  shew^,  without  reference  to 
the  mnemonic  verses,  that  if  the  minor  premisses  are 
contradictories,  the  conclusions  will  not  be  contra- 
dictories. 

284.  Is  it  possible  that  there  should  be  a  valid 
syllogism  such  that,  each  of  the  premisses  being  con- 
verted, a  new  syllogism  is  obtainable  giving  a  conclu- 
sion in  which  the  old  major  and  minor  terms  have 
changed  places  t 

Prove  the  correctness  of  your  answer  by  general 
reasoning,  and  if  it  is  in  the  affirmative,  determine 
the  syllogism  or  syllogisms  fulfilling  the  given  con- 
ditions. 

If  such  a  syllogism  is  possible,  it  cannot  have  two  afl^r- 
mative  premisses,  or  (since  A  can  only  be  converted  per 


288 


SYLLOGISMS. 


[part  III. 


CHAP.  XI.] 


SYLLOGISMS. 


289 


accidens)  we  should  have  two  particular  premisses  in  the 
new  syllogism. 

Therefore,  the  original  syllogism  must  have  one  negative 
premiss.     This  cannot  be  O,  since  O  is  inconvertible. 

Therefore,  one  premiss  of  the  original  syllogism  must  be  B. 

First,  let  this  be  the  major  premiss.  Then  the  minor 
premiss  must  be  affirmative,  and  its  converse  being  a  par- 
ticular affirmative  will  not  distribute  either  of  its  terms.  But 
this  converse  will  be  the  major  premiss  of  the  new  syllogism, 
which  also  must  have  a  negative  conclusion.  We  should 
then  have  illicit  major  in  the  new  syllogism,  and  this  suppo- 
sition will  not  give  us  the  desired  result 

Secondly,  let  the  minor  premiss  of  the  original  syllogism 
be  E.  The  major  premiss  in  order  to  distribute  the  old 
major  term  must  be  A,  with  the  major  term  as  subject. 
We  get  then  the  following,  satisfying  the  given  conditions : — 

All  F  is  M, 
No  Mis  5,  or  No  5  is  M, 

therefore,         No  6'  is  F,  or  Some  6"  is  not  F. 

that  is,  we  really  have  four  syllogisms,  such  that  both  pre- 
misses being  converted,  thus, — 

No^'isiT/,  or  Noil/ is  6", 
Some  M  is  F, — 

we  have  a  new  syllogism  giving  a  conclusion  in  which  the 
old  major  and  minor  terms  have  changed  places,  namely. 

Some  F  is  not  S. 

Symbolically, — 


FaM, 
MeSA 
or  SeM,] 

.'.  SeF) 
or  SoF) 


SeMA 
or  AfeS,} 
MiF, 

.-.  FoS. 


If  it  had  been  required  to  retain  the  quantity  of  the 
original  conclusion,  this  must  be  SoF,  so  that  we  should 
have  only  two  syllogisms  fulfilling  the  given  conditions. 

285.  Ls  it  possible  that  there  should  be  two 
syllogisms  having  a  common  premiss  such  that  their 
conclusions,  being  combined  as  premisses  in  a  new 
syllogism,  may  give  a  universal  conclusion  ?  If  so, 
determine  what  the  two  syllogisms  must  be.       [n.] 


;*'<■ 
f,  '' 


K.  L. 


19 


PART    IV. 

A  GENERALISATION  OF  LOGICAL  PRO- 
CESSES IN  2 HEIR  APPLICATION  10  COM- 
FLEX  PROPOSITIONS. 

CHAPTER  I. 

THE   COxMBINATION   OF   SIMPLE   TERiMS. 


286.     Complex  Terms. 

A  simple  term  may  for  our  present  purpose  be  defined 
as  one  which  is  represented  by  a  single  symbol .;  e.^.^ 
Ay  P,  X.  The  combination  of  simple  terms  yields  a  com- 
plex term. 

Simple  terms  may  be  combined  (i)  conjunctively,  or 
(2)  disjunctively. 

(i)  "What  is  both  A  and  B''  is  a  complex  term  result- 
ing from  the  conjunctive  combination  of  the  simple  terms  A 
and  B^,  It  is  convenient  to  denote  a  complex  term  of  this 
kind  by  a  simple  juxtaposition  of  the  terms  involved,  thus — 

1  This  species  of  complex  term  is  called  by  Jevons  a  combined  term 
{Pure  Logic,  p.  15).  So  far  as  it  requires  a  distinctive  name  I  think 
I  should  prefer  to  call  it  a  conjunctive  term. 


CHAP.  I.] 


COMPLEX    INFERENCES. 


291 


AB.     Accordingly  the  proposition  ^^  AB  is  CD  "  would  be 
read  "  Anything  that  is  both  A  and  B  is  both  C  and  Z>." 

(2)  "What  is  either  ^  or  ^*'  is  a  complex  term  result- 
ing from  the  disjunctive  combination  of  the  simple  terms  A 
and^^ 

In  what  follows  it  must  be  remembered  that  I  have 
adopted  the  view,  that  logically  the  alternatives  in  a  dis- 
junction, (unless  they  are  formal  contradictories),  are  non- 
exclusive. Thus,  if  we  speak  of  anything  as  being  "  A  or 
B "  we  do  not  exclude  the  possibility  of  its  being  both  A 
and  By  (compare  section  109).  In  other  words  "^  or  B" 
does  not  exclude  "  AB." 

The  force  of  a  disjunctive  term  when  it  is  the  subject  of 
a  proposition  should  be  carefully  noted.  "Anything  that  is 
either  Pot  Q  is  ^,"  or  "whatever  is  either  P  or  Q  is  R,^^ 
may  sometimes  for  the  sake  of  brevity  be  written  "/*or  Q 
is  R.'*  The  latter  expression,  however,  might  also  be  inter- 
preted to  mean  "one  of  the  two  P  or  Q  is  R,  but  we  do 
not  know  which";  and  in  consequence  of  this  possible  am- 
biguity, the  more  definite  mode  of  statement,  "Whatever 
is  either  P  or  Qis  R^"  is  to  be  preferred. 

A  complex  term  may  of  course  involve  both  conjunctive 
and  disjunctive  combination :  e.g.y  "  AB  or  CD"  It  is  to 
be  noted  that  the  statement  that  anything  is  "A  or  B  and 


^  This  kind  of  complex  term  is  called  by  Jevons  a  plural  term 
{Pure  Logic,  p.  25).  So  far  as  it  requires  a  distinctive  name  I  think 
I  should  prefer  to  call  it  a  disjunctive  term. 

2  The  subject  of  this  proposition  is  to  be  regarded  as  a  single  dis- 
junctive term.  The  same  meaning  might  be  given  by  saying  "/*  and 
Q  are  A\"  but  in  this  case  I  should  consider  that  we  have  two  distinct 
subjects,  and  two  propositions  eUiptically  expressed. 

19 2 


292 


COMPLEX   INFERENCES. 


[part  IV. 


at  the  same  time  C  or  Z> "  is  equivalent  to  the  statement 
that  it  is  ''A  Cor  AD  or  BC  or  BD:' 

We  speak  of  ?i  propositio?i  as  being  complex  if  either  its 
subject  or  its  predicate  is  a  complex  term. 

287.  In  a  complex  term  the  order  of  combination 
is  indifferent. 

This  is  true  whether  the  combination  be  conjunctive  or 
disjunctive. 

Thus,  AB  and  BA  are  precisely  the  same  terms.  It  is 
obviously  the  same  thing  if  we  speak  of  anything  as  being 
both  A  and  B,  or  if  we  speak  of  it  as  being  both  B  and  A, 

Again  "^  or  B"  and  ^^B  or  ^"  have  precisely  the  same 
signification.  It  is  the  same  thing  to  speak  of  anything  as 
being  A  or  B  as  to  speak  of  it  as  being  B  or  A. 

288.  The  Opposition  of  Complex  Terms. 

We  shall  find  it  convenient  to  denote  the  contradictory 
of  any  simple  term  by  the  corresponding  small  letter.  Thus 
for  not-^  we  write  a^  for  not-^  we  write  b.  A  and  a  there- 
fore denote  between  them  the  whole  universe  of  discourse 
(whatever  that  may  be),  but  they  denote  nothing  in  common. 
In  other  words,  whatever  A  may  designate,  it  is  necessarily 
true  that  Everything  (in  the  universe  of  discourse)  is  ^  or  ^; 
and  that  A  is  not  a.  It  also  follows  that  Aa  necessarily 
represents  a  non-existent  class ;  what  is  both  A  and  not--^ 
cannot  have  a  place  in  any  universe. 

However  complex  a  term  may  be,  we  can  always  find  its 
contradictory  by  applying  the  criterion  laid  down  in  section 
28.  "A  pair  of  contradictory  terms  are  so  related  that 
between  them  they  exhaust  the  entire  universe  to  which 
reference  is  made,  whilst  there  is  no  individual  of  which  they 
can  both  be  at  the  same  time  affirmed." 


CHAP.  I.] 


COMPLEX   INFERENCES. 


293 


{: 


Now  whatever  is  not  AB  must  be  either  a  or  b,  whilst 
nothing  that  is  AB  can  be  either  of  these;  and  vice  versa. 

(AB, 
\a  or  b, 

are  therefore  a  pair  of  contradictories. 

Similarly, 

A  or  B, 

:ab, 

are  a  pair  of  contradictories. 

If,  then,  two  simple  terms  are  conjunctively  combined 
into  a  complex  term,  the  contradictory  of  this  complex  term 
is  given  by  disjunctively  combining  the  contradictories  of  the 
simple  terms.  And,  conversely,  if  two  simple  terms  are  dis- 
junctively combined  into  a  complex  term,  the  contradictory 
of  this  complex  term  is  given  by  conjunctively  combining 
the  contradictories  of  the  simple  terms. 

In  each  case,  we  substitute  for  the  simple  terms  involved 
their  contradictories,  and  (as  the  case  may  be)  change  and 
for  or,  or  or  for  and. 

But  however  complex  a  term  may  be,  it  must  consist  of 

a  series  of  conjunctive  and  disjunctive  combinations,  and  it 

may  be  successively  resolved  into  the  combination  of  pairs 

of  relatively  simple  terms  till  it  is  at  last  shewn  to  result 

from  the   combination   of  absolutely  simple   terms.      For 

example, — 

ABC  or  BE  ox  FG 

results  from  the  disjunctive  combination  of  the  pair, — 

iABC  or  DE, 
\       FG', 

ABC  or  DE  results  from  the  disjunctive  combination  of 
the  pair, — 

iABC, 

WE', 


294  COMPLEX   INFERENCES.  [part  IV. 

FG  results  from  the  conjunctive  combination  of  the  pair, — 

KG; 

and  similarly  we  may  resolve  ABC,  DE. 

We  may  hence  deduce  the  following  general  rule  for 
obtaining  the  contradictory  of  any  complex  term : — For  each 
simple  term  involved,  substitute  its  contradictory ;  everywJiere 
change  and  for  or,  a7id  or  for  and  \  This  rule  is  of  simple 
application,  and  in  what  follows  will  be  found  to  be  of  very 
considerable  importance.  Its  full  force  will  be  made  more 
apparent  later  on. 

Thus  the  contradictory  of 

A  or  BC 
is    a  and  {p  or  r), 
i.e.y     ab  or  ac. 

The  contradictory  of 

ABC  or  ABD 
is     {a  or  b  or  c)  and  (a  or  b  or  d), 

which,  as  we  shall  presently  shew^  is  resolvable  into 

a  Q\  b  01  cd. 

In  such  statements  as  the  above,  the  use  of  brackets  is 
necessary  to  avoid  ambiguity.  Thus,  ^  or  ^  or  ^  and  a  or  b 
or  d  might  be  read  a  or  b  ox  ca  or  b  ox  d\  but  we  really 
mean  that  each  term  in  the  second  set  of  alternatives  is  to 
be  conjunctively  combined  with  each  term  in  the  first  set  of 
alternatives. 

1  In  applying  this  rule,  the  information  given  by  two  such  proposi- 
tions as  **A'  is  /•,"  "  F  is  P,"  if  stated  in  the  form  of  a  single  pro- 
position, must  be  expressed  "  What  is  either  X  oi  F  is  /*,"  not  "  X  and 

Fare  /*."     Compare  section  286. 

2  Cf.  section  196. 


CHAP.  I.J 


COMPLEX   INFERENCES. 


295 


Two  terms  may  be  inconsistent  v/ithout  being  contra- 
dictories; i.e.,  they  cannot  both  be  affirmed  of  anything, 
but  it  may  be  that  there  are  some  things  of  which  neither  can 
be  affirmed.  Thus,  we  can  say  that,  whatever  A,  B  and  C 
may  stand  for,  ''AB  is  not  bC^'  (since  if  AB  were  bC  \\. 
would  involve  something  being  at  the  same  time  both  B 
and  not-^) ;  but  we  cannot  say  that,  whatever  A,  B  and  C 
may  stand  for,  "  Everything  is  AB  or  ^C,"  (since  something 
might  be  Abe,  which  is  neither  AB  nor  bC).  If  a  con- 
junctive term  contains  a  term  which  is  the  contradictory  of 
a  term  contained  in  another  conjunctive  term  then  it  follows 
that  these  two  conjunctive  terms  are  inconsistent. 

If  two  conjunctive  terms  are  such  that  every  term  in 
one  has  corresponding  to  it  in  the  other  its  contradictory, 
these  two  terms  may  be  regarded  as  logical  contraries,  (com- 
l)are  the  definition  of  contrary  terms  given  in  section  28). 
Thus,  AbC,  aBc  may  be  spoken  of  as  contraries. 

289.     The  Development  of  Terms  by  means  of 
the  Law  of  Excluded  Middle. 
By  the  Law  of  Excluded  Middle, 

Everthing  is  B  ox  b, 

and  therefore,  A  is  AB  or  Ab. 

Again,        Everything  is  C  ox  c; 
therefore,      AB  is  ABC  or  ABc, 
and  Ab  is  AbC  or  Abe, 

therefore,  A  is  ABC  or  ABc  or  AbC  or  Abe. 

This  is  called  the  development  of  a  term  with  reference 
to  other  terms ;  thus,  A  is  here  developed  with  reference  to 
B  and  C.  Compare  Jevons,  Pure  Logic,  p.  37.  He  calls 
any  two  alternatives  which  are  the  same,  except  as  regards 
one  term  in  each  which  are  contradictories,  a  dual  term. 
Thus,  ''AB  or  Ab''  is  a  dual  term  as  regards  B. 


CHAPTER   11. 

THE   SIMPLIFICATION    OF    COMPLEX    PROPOSITIONS. 

290.     Types  of  Complex  Propositions. 

Complex  Propositions  may  be  divided, — 

First,  (as  in  the  case  of  simple  propositions),  according 
as  they  are  affirmative  or  negative ; 

e.g..       All  AB  is  C  or  Z); 
No  AB  is  C  or  Z>. 

Secondly,  (also  as  in  the  case  of  simple  propositions), 
according  as  they  are  universal  or  particular ; 

e.g.,  All  AB  is  C  or  B ; 
Some  AB  is  C  or  Z>. 

We  shall  deal  very  little  with  particular  complex  propo- 
sitions, and  it  will  frequently  be  found  convenient  to  write 
universal  complex  propositions  in  the  indefinite  form.  Thus, 
by  AB  is  C  or  D  we  understand, — 

All  AB  is  C  or  Z>. 

Thirdly,  according  as  only  the  subject  or  only  the 
predicate  or  both  subject  and  predicate  are  complex  terms  ; 

e.g.,      AB  is  C, 

A  is  B  or  C, 
AB  is  C  or  D. 


CHAP.  II.]  COMPLEX    INFERENCES.  297 

Fourthly,  according  as  there  is  or  is  not 

(a)     conjunctive  combination  in  the  subject, — 

e.g.,     ^  is  Cor  D, 

AB'i?.  Cor  £>', 

{P)     conjunctive  combination  in  the  predicate, — 

e.g.,     AB  is  C, 
ABis  CD\ 

(y)     disjunctive  combination  in  the  subject, — 

e.g.,     A  is  CD, 
Whatever  is  either  ^  or  ^  is  CD ; 

(8)     disjunctive  combination  in  the  predicate, — 

e.g.,     AB  is  C, 

AB  is  Cor  D. 

291.  The  Resolution  of  Complex  into  relatively 
Simple  Propositions. 

Affirmative.  Affirmative  complex  propositions  may  be 
immediately  resolved  into  relatively  simple  ones,  so  far  as 
there  is  conjunctive  combination  in  the  predicate,  or  dis- 
junctive combination  in  the  subject.     Thus, — 

(i)  X'lsAB 

is  obviously  resolvable  into  the  two  propositions, — 

Xis^, 
XxsB. 

(2)     Whatever  is  either  Jf  or  Fis  ^, 

is  obviously  resolvable  into  the  two  propositions, — 

X'xsA, 
Y\^A, 


298 


COMPLEX   INFERENCES. 


[part  IV. 


Negative.  Negative  complex  propositions  may  be  im- 
mediately resolved  into  relatively  simple  ones,  so  far  as  there 
is  disjunctive  combination  either  in  the  subject  or  in  the 
predicate.     Thus, 

(i)     Nothing  that  is  either  X  01  Y\^  A 
is  obviously  resolvable  into, — 

No  X  is  A, 
No  Y  is  A, 

(2)  No  X  is  ^  or  ^ 

is  obviously  resolvable  into, — 

NoXis^, 
No  X  is  B, 

The  difference  between  affirmative  and  negative  proposi- 
tions here  must  be  carefully  noticed.  So  far  as  there  is  con- 
junctive combination  in  the  subject  or  disjunctive  combina- 
tion in  the  predicate  of  an  affirmative  proposition,  or  con- 
junctive combination  either  in  the  subject  or  in  the  predicate 
of  a  negative  proposition,  we  cannot  immediately  resolve  it 
into  simpler  propositions. 

Even  in  these  cases,  however,  complex  propositions 
may  be  resolved  into  relatively  simple  ones  in  a  more 
roundabout  way,  namely,  by  the  aid  of  obversion  or  con- 
traposition, as  will  be  shewn  subsequently.  Compare  espe- 
cially chapter  v. 

292.     The  Equivalence  of  Propositions. 

Two  propositions  are  equivalent  if  each  can  be  inferred 
from  the  other.  Similarly,  two  sets  of  propositions  are 
equivalent  if  every  member  of  each  set  can  be  inferred  from 
the  other  set. 


CHAP.  II.] 


COMPLEX   INFERENCES. 


299 


When  we  omit  terms  from  a  proposition,  or  introduce 
fresh  terms,  or  when  in  any  way  we  obtain  a  proposition  or 
set  of  propositions  from  another  proposition  or  set  of  pro- 
positions, we  should  carefully  distinguish  two  cases: — 

First,  where  the  force  of  the  original  statement  is  un- 
affected, so  that  we  can  pass  back  from  the  new  proposi- 
tion or  propositions  to  the  original  proposition  or  proposi- 
tions. 

Secondly,  where  the  force  of  the  original  statement  is 
weakened,  so  that  we  cannot  pass  back  from  the  new  pro- 
position or  propositions  to  the  original  proposition  or  pro- 
positions. 

In  many  cases  it  is  of  very  great  importance  to  know 
whether  in  a  process  of  manipulation  we  have  or  have  not 
lost  any  of  the  information  originally  given  us. 

293.  The  Omission  of  Terms  from  a  Complex 
Proposition,  the  force  of  the  assertion  remaining  unaf- 
fected. 

(i)  //  is  superfluous  for  any  simple  term  to  appear  more 
than  once  in  a  conjunctive  term. 

Thus  A  A  merely  denotes  the  class  A,  ABB  merely 
denotes  the  class  AB.  Such  terms  in  their  original  form 
are  tautologous,  and  the  repetition  of  the  term  should 
therefore  be  struck  out.  Compare  Boole,  Laws  of  Thought^ 
p.  31,  and  Jevons,  Pure  Logic,  p.  15. 

(2)  Ln  a  series  of  alternatives  it  is  superfluous  for  any 
given  alternative  to  be  repeated. 

To  say  that  anything  is  **^  or  A''  is  to  say  that  it  is  ^; 
to  say  that  anything  is  ^^  A  or  BC  or  BC^^  is  to  say  that  it 
is  *'-^  or  BC",     The  repetition  of  an  alternative  should 


?oo 


COMPLEX   INFERENCES. 


[part  IV. 


therefore  always  be  struck  out.    Compare  Jevons,  Pure  Logic, 
p.  26. 

(3)  In  a  universal  negative  proposition  it  is  superfluous 
for  the  same  term  to  appear  in  every  alternative  in  the  subject 
and  also  in  every  alternative  in  the  predicate,  that  is,  in  such 
a  case  it  may  be  otnitted  either  from  the  subject  or  from  the 

predicate. 

For  example,  to  say  that  No  AB  is  ^Cis  precisely  the 
same  as  to  say  that  No  AB  is  C,  or  that  No  B  is  AC.  For 
to  say  that  No  AB  is  AC  is  the  same  thing  as  to  deny  that 
anything  is  ABAC',  but,  as  shewn  above,  the  repetition  of 
the  term  A  is  superfluous,  and  the  statement  may  therefore 
be  reduced  to  the  denial  that  anything  is  ABC.  And  this 
may  equally  well  be  expressed  by  saying  No  AB  is  C,  or 
No  B  is  AC.  Compare  also  Chapter  iii,  On  the  Conversion 
of  Complex  Propositions. 

Similarly,  No  AB  is  ^C  or  AD  may  be  reduced  to  No 
AB  is  C  or  D,  or  to  No  j9  is  ^C or  AB. 

(4)  If  in  an  affirmative  proposition  a  term  that  appears 
in  every  alternative  in  the  subject  appears  also  in  any  alterna- 
tive in  the  predicate,  it  may  be  dropped  from  the  latter  without 
affecting  the  force  of  the  state?nent. 

A  is  AB 
may,  (as  shewn  in  section  291),  be  resolved  into 

A  is  A, 
AisB. 

But  A  is  A  is  di.  merely  identical  proposition  and  gives 
no  information. 

^is^^ 
may  therefore  be  reduced  to  the  single  proposition 

^is^. 


CHAP.  II.]  COMPLEX   INFERENCES.  301 

Similarly, 

^^is^Cor^C 

may  be  reduced  to 

AB  is  C  or  C, 

and  therefore,  as  shewn  above,  to 

AB  is  C 

(5)  ^f  ^^^^  ^f  ^  ^^^^^^  of  alternatives  is  merely  a  subdivi- 
sion of  another  of  the  alternatives  it  may  be  ojnitted  without 
destroying  any  of  the  force  of  the  original  assertion.  In  other 
words,  in  a  disjunctive  term,  "any  alternative  may  be  re- 
moved, of  which  a  part  forms  another  alternative,"  (com- 
pare Jevons,  Pure  Logic,  p.  26). 

Thus,  AB  is  a  subdivision  of  A,  and  ''A  or  AB''  may 
therefore  be  reduced  to  "^." 

This  may  be  shewn  as  follows  : — 

Xis  A  ox  AB; 

but,   AB  is  A, 

therefore,   X  is  ^  or  A, 

therefore,   ^  is  ^ ; 

and  conversely,  if  X  is  A,  since,  by  the  law  of  excluded 

middle,  A  is  AB  or  Ab,  it  follows  that  X  is  Ab  or  AB; 

but,   Ab  is  A, 
therefore,    X  is  ^  or  AB. 

Similarly, 

X  is  AB  or  ABC  or  BD 

may  be  simplified  by  the  omission  oi  ABC,  becoming 

X  is  AB  or  BD. 

(6)  Any  term  which  represetits  a  non-existent  class  may 
obviously  be  dropped  from  a  series  of  alternatives  without 
altering  the  force  of  the  proposition ;  this  is  the  case  with 


302 


COMPLEX   INFERENCES. 


[part  IV. 


a  term  which  involves  a  self-contradiction.  "  Aa "  means 
that  which  is  both  A  and  not-^,  but  by  the  law  of  Contra- 
diction, no  such  class  is  possible.  Such  a  term  as  Aa  may 
therefore  always  be  dropped.  It  follows,  therefore,  that  if 
Jf  is  -^  or  Bbj  X\s  A ;  and  it  is  clear  that  there  is  here  no 
weakening  of  the  force  of  the  original  proposition. 

(7)  Reductioji  of  dual  terms,     (Compare  section  289.) 

Another  simplification  is  possible  where  we  have  two 
alternatives,  one  of  which  contains  a  term  which  is  the 
contradictory  of  a  term  contained  in  the  other,  the  remain- 
ing terms  in  each  being  the  same;  c.g,^  ABC  ox  ABc.  These 
may  be  replaced  by  a  single  term  containing  only  the  ele- 
ments which  are  common  to  both  the  original  alternatives, 
without  any  of  the  force  of  the  original  proposition  being 
lost.  Thus,  ''ABC  or  ABc''  may  be  replaced  by  ''ABr 
For  ABC  and  ABc  are  both  AB  \  and  conversely,  by  the 
law  of  excluded  middle,  AB  is  ABC  or  ABc. 

Thus,  X  is  AB  or  Ab ; 

but,  AB  is  A,  and  Ab  is  A  ; 
therefore,  X  is  A. 

We  have  also,  X  is  A  ; 

but,  A  is  AB  or  Aby 
therefore,  X  is  AB  or  Ab. 

(8)  //  in  a  series  of  alternatives  occurring  either  in  the 
subject  or  in  the  predicate  of  a  proposition^  the  contradictory  of 
any  given  alternative  appears  combined  with  other  terms  in 
other  alternatives  it  may  be  omitted  from  the  latter  without 
altering  the  force  of  the  assertion. 

Thus,  "  A  or  aB  "  may  be  replaced  by  "  A  or  .5,"  and 
vice  versa. 


CHAP.  II.] 

For, 


COMPLEX   INFERENCES. 


303 


given  X  is  ^  or  aB ; 

since  aB  is  B^ 

it  follows  that  X  is  ^  or  B. 

And,  conversely,  given  ^  is  ^  or  ^ ; 

since  B  is  AB  or  aB^ 
it  follows  that  X  is  ^  or  AB  or  aB  ; 
but,  AB  is  A, 
therefore,  X  is  ^  or  aB. 

Thus,  we  may  not  merely  infer 

''X  is  A  or  ^"  from  "X  is  A  or  aB'') 
but  we  may  do  this  without  any  loss  of  force. 
Again,  given  No  X  is  either  A  ox  aB  \ 

since  AB  is  A, 

it  follows  that  X  is  not  AB  ; 

therefore,  No  X  is  either  AB  or  aB  \ 

but  B  is  either  AB  or  aB, 

therefore,  X  is  not  B ; 

and  therefore,  No  X  is  either  A  or  B. 

In  this  case  the  passage  back  from 

No  X  is  either  A  or  B, 
to  No  X  is  either  A  or  aB, 

is  still  more  obvious. 

294.  The  Introduction  of  fresh  Terms  into  Com- 
plex Propositions  without  affecting  the  force  of  the 
assertion. 

This  is  in  itself  the  reverse  of  simplification,  but  it  is  in 
some  cases  a  necessary  introduction  to  a  process  of  sim- 
plification. 

It  is  clear  that  we  have  a  case  corresponding  to  each  of 
the  cases  just  discussed.  Wherever  we  may  obtain  a  pro- 
position equivalent  to  a  given  proposition  by  dropping  a 


304 


COMPLEX    INFERENCES. 


[part  IV. 


term,  we  may  correspondingly  obtain  a  proposition  equi- 
valent to  a  given  proposition  by  the  introduction  of  a  fresh 
term.  The  proof  of  each  separate  case  has  been  given 
in  establishing  the  various  equivalences  in  the  preceding 
section. 

The  following  are  the  more  important  cases : 

(3)  In  a  universal  negative  proposition  any  term  that 
appears  in  ei'ery  alternative  in  the  subject  may  be  combined 
with  any  alternative  in  the  predicate ;  similarly^  any  term  that 
appears  in  every  alternative  in  the  predicate  may  be  combined 
with  afiy  alternative  in  the  subject ;  and  in  neitfier  case  will 
the  force  of  the  original  statement  be  ajfected. 

(4)  /;/  an  affirmative  proposition  any  terfn  that  appears 
in  ei'ery  alternative  in  the  subject  may  be  coinbined  with  any 
alternative  in  the  predicate  ^vithout  affecting  the  force  of  the 
statement, 

(7)  If  any  term  either  in  the  sidject  or  in  the  predicate 
of  a  proposition  is  developed  by  meatis  of  the  law  of  excluded 
middle  (cf.  section  289)  7i.te  obtain  an  equivalent  proposition, 

(8)  In  a  series  of  alternatives  occurring  either  in    the 
subject  or  in  the  predicate  of  a  proposition^  the  contradictory  of 
any  given  alternative  may  be  combined  luith  other  alternatives 
without  altering  the  force  of  the  assertion.     For  example, 
"^  or  aB''  may  be  substituted  for  ''A  or  B'' 

295.  Types  of  Equivalent  Propositions,  as  esta- 
blished in  the  two  preceding  sections. 


(i)     lX  is  ABB\ 
i^  is  AB. 


(2) 


{ 


X  is  AB  or  AB-, 
X  is  AB. 


CHAP.  II.]  COMPLEX   INFERENCES. 


305 


(3) 

(4) 
(5) 
(6) 

(7) 
(8) 


^o  AX\%AB  ox  ACD\ 
No  ^X  is  ^  or  CD', 
ISio  X  is  AB  or  A  CB. 

A  is  AB  or  ACB; 
A  is  B  or  CD. 

X  is  A  or  B  or  BC; 
X  is  ^  or  B. 

X  is  A  or  Bb; 
X  is  A. 

X  is  ABC  or  ABc; 
X  is  AB. 

X  is  ^  or  aB; 
X  is  ^  or  B. 


296.  Shew  that  "a  or  b  or  cd''  is  the  contra- 
dictory of  "ABC  or  ABDr 

A  proof  of  this  was  promised  in  section  288.  Applying 
the  rule  laid  down  in  that  section,  we  have  for  the  contra- 
dictory of  the  given  term, — 

{a  or  b  or  c)  and  at  the  same  time  {a  or  b  or  d). 

We  have  therefore  to  combine  each  of  the  first  set  of  alter- 
natives with  each  of  the  second  set.     This  yields 

aa  or  ab  or  ad  or  ab  or  bb  or  bd  or  ac  or  be  or  cd. 

But  we  have  shewn  that  aa  may  be  replaced  by  a,  and 
bb  by  b ;  that  since  ab,  ad,  and  ac  are  subdivisions  of  a,  and 
a  is  one  of  the  alternatives,  they  may  be  omitted;  and 
similarly  with  bd  and  be,  in  consequence  of  their  relation 
to^. 

The  contradictory  is  therefore  reduced  to  a  or  b  or  cd. 

K.  L.  20 


3o6 


COMPLEX   INFERENCES.  [part  iv. 


297.  State  the  contradictories  of  the  following 
terms  in  their  simplest  forms  : — 

AB  or  BC  or  CD, 
AB  or  bC   or  cD, 
ab    ox  BC  or  cd, 
AB  or  bC  or  Cd, 

298.  Shew  that  the  two  following  propositions 
are  equivalent  to  each  other : — 

(i)     X  \s  BC  or  bD  OY  CD, 
(2)     X  is  BC  or  bD. 

299.  Shew  the  equivalence  between  the  pro- 
positions,— 

(i)  ^F  is  either  aB  or  aC  or  bC  or  aE  or  bE 
or  Ad  or  Ae  or  bd  or  <^^  or  r(/  or  ce ; 
(2)  X  F  is  either  rt:  or  ^  or  ^  or  e. 
(2)  follows  immediately  from  (i);  but  it  is  important  to 
notice  also  that  nothing  is  lost  in  this  inference,  />.,that  we 
may  pass  back  from  (2)  to  (i).  This  follows  immediately 
from  the  principles  established  in  sections  293 — 295. 

The  steps  may  be  shewn  at  length  as  follows,  (the 
numbers  indicating  the  processes  made  use  of  as  described 
in  the  above  sections).  A  comma  is  here  placed  between 
the  different  alternatives. 

a,  b,  d,  e ; 

ay  b,  d,  cd,  e,  ce;  (5) 

a,  b,  Ad,  cd,  Ae,  ce\  (8) 

a,  aC,  aE,  b.  Ad,  cd,  Ae,  ce;  (5) 

aB,  aC,  aE,  b,  Ad,  cd,  Ae,  ce;  (8) 

aB,  aC,  aE,  b,  bC,  bd,  Ad,  cd,  Ae,  ce;       (5) 

aB,  aC,  aE,  bE,  be,  bC,  bd.  Ad,  cd,  Ae,  ce.    (7) 


CHAP.  II.]  COMPLEX    INFERENCES.  307 

300.  Shew  the  equivalence  between  the  two  pro- 
positions : — 

(i)  XF  is  aBC  or  aCD  or  aBe  or  aDe  or  AcD 
or  abD  or  bcD  or  aDE  or  cDE ; 

(2)     X  F  is  aBC  or  aD  or  cD  or  aBe. 

301.  Shew  the  equivalence  between  each  pair  of 
the  following  propositions  : — 

(i)  X  is  either  AB  or  AC  or  BC  or  abc  or  aB 
or  C; 

X  is  either  a  or  B  or  C. 

(ii)  Xis  either  aBC  or  aBd  or  acd  or  bed  or  ABd 
or  Acd  or  abd  or  aCd  or  BCd; 

X  is  either  Bd  or  cd  or  ad  or  aBC  or  aCd. 

(in)  X  is  either  Pqr  or  pQs  or  prs  or  (^rs  or  pg 
or  />S  or  gR  ; 

X  is  either/  or  g. 

302.  Shew  the  equivalence  between  the  two  pro- 
positions : — 

No  X  F  is  either  aB  or  aC  or  bC  or  aE  or  bE 
or  Ad  or  ^4^  or  ^^  or  be  or  ^^  or  ce  ; 
No  X  F  is  either  a  or  b  or  d  or  ^. 

Here  it  is  immediately  obvious  that  the  first  proposition 
is  inferrible  from  the  second.  It  may  be  shewn  that  the 
second  is  inferrible  from  the  first  by  a  kind  of  converse 
process  to  that  employed  in  section  299. 

303.  Shew  the  equivalence  between  the  two  pro- 
positions : — 

No  XF  is  aBC  or  aCD  or  aBe  or  aDe  or  AcD  or 
abD  or  bcD  or  aDE  or  cDE  ; 

20 — 2 


3o8 


COMPLEX    INFERENCES. 


[part  IV. 


No  X  V  is  aBC  or  aD  or  cB  or  aBe. 

No  difficulty  will  be  found  with  this  example  if  the 
student  notices  that  "neither  Ac£>  nor  aZ>''  may  be  re- 
duced to  "  neither  cB  nor  aZ>,''  since  cD  must  be  either 
AcD  or  aD. 

304.  Shew  the  equivalence  between  the  following 
pairs  of  propositions : — 

(i)  No  X  is  either  AB  or  AC  or  BC  or  adc 
or  aB  or  C ; 

No  X  is  either  a  or  B  or  C, 

(ii)  No  X  is  either  ^^C  or  aBd  or  ^r^  or  bed 
or  -^^r/  or  y^a/  or  abd  or  ^C^  or  BCd\ 

No  JiT  is  either  Bd  or  ^^  or  ^^  or  aBC  or  ^C<3^. 

(iii)  No  X  is  either  P^r  or  pQs  or  /r^-  or  qrs 
or  pq  or  pS  or  ^7?; 

No  X  is  either/  or  q, 

305.  Simplify  the  propositions  : — 

(i)     ^  is  Ab  or  aC  or  7?(7^  or  Be  or  ^Z^  or  6"/?. 
(2)     X  is  y^  CZ)  or  Ae  or  ^^  or  aB  or  ^CZ^. 

306.  Inference  by  the  Omission  of  Terms,  or 
by  the  Introduction  of  fresh  Terms  in  Complex  Pro- 
positions, the  inferred  proposition,  however,  not  being 
equivalent  to  the  original  proposition. 

(i)  A  fresh  term  may  always  be  introduced  into  the 
subject  of  a  proposition,  (though  the  force  of  the  proposition 
is  thereby  weakened);  but  no  term  may  ever  be  omitted 
from  the  subject  of  a  proposition,  (except  in  the  case  of  a 
negative  proposition  where  the  same  term  appears  also  in 
the  predicate  as  shewn  in  section  293). 


CHAP.  II.] 


COMPLEX   INFERENCES. 


309 


It  is  clear  that  whatever  may  be  affirmed  (or  denied)  of 
A  may  be  affirmed  (or  denied)  of  AB  \  in  other  words, 
whatever  is  true  of  A  is  true  of  that  which  is  both  A  and  B. 

But  we  cannot  on  the  other  hand  pass  back  from  affirm- 
ing (or  denying)  anything  of  AB  to  affirming  (or  denying) 
the  same  thing  of  A. 

(2)  A  fresh  term  may  always  be  introduced  into  the 
predicate  of  a  negative  proposition;  but  not  into  the  predi- 
cate of  an  affirmative  proposition,  unless  it  already  appears 
in  the  subject  \ 

(3)  A  term  may  always  be  dropped  from  the  predicate 
of  an  affirmative  proposition;  but  not  from  the  predi- 
cate of  a  negative  proposition,  unless  it  also  appears  in  the 
subject  ^ 

It  is  clear  that  if  No  A  is  B,  then  No  A  is  both 
B  and  C ;  but  not  vice  versa^  since  although  No  A  is  both 
B  and  C,  All  A  might  be  B  and  not-  C.  Again,  it  is  clear 
that  if  All  A  is  both  B  and  C,  then  All  A  is  B ;  but  it 
does  not  follow  that  if  All  A  is  B,  therefore  All  A  is  both 
B  and  C, 

307.  If  no  ^  is  be  or  Cd,  it  follows  that  no  A 
is  bd. 

308.  Interpretation  of  propositions  of  the  forms 
No  AB  is  B,  AB  is  a,  AB  is  Ce. 

Propositions  of  the  above  kind  may  easily  result  as  a 
consequence  of  the  manipulation  of  complex  propositions ; 
but  they  involve  a  contradiction  in  terms  and  are  in  direct 
contravention  of  the  fundamental  laws  of  thought.  They 
must  be  interpreted  as  affirming  the  non-existence  of  the 

^  Cf.  sections  293,  294. 


3IO 


COMPLEX   INFERENCES. 


[part  IV. 


subject  of  the  proposition.     Thus,  AB  is  a  is  to  be  inter- 
preted No  A  is  B,  or  A  is  b. 

This  must  be  taken  in  connection  with  the  discussion 
in  section  io6.  The  view  was  there  adopted  that  no  uni- 
versal proposition  implies  the  existence  of  its  subject;  but 
if  it  is  affirmative  it  denies  the  existence  of  anything  that 
is  the  subject  and  is  not  the  predicate.  Thus  AB  is  a 
denies  the  existence  of  anything  that  is  at  the  same  time 
AB  and  not-«,  i.e.,  A.  But  AB  is  AB  and  A.  The 
existence  of  AB  is  therefore  denied. 

Similarly,  a  universal  negative  proposition  denies  the 
existence  of  anything  that  is  both  subject  and  predicate. 
No  AB  is  B  denies  the  existence  of  ABB,  i.e.,  of  AB 
as  before. 

AB  is  Cc  affirms  that  AB  is  something  that  is  non- 
existent, and  therefore  that  it  is  itself  non-existent. 

If  the  view  were  adopted  that  a  proposition  does  imply 
the  existence  of  its  subject,  then  if  propositions  of  the 
above  form  were  obtained,  we  should  be  thrown  back  on 
the  alternative  that  some  inconsistency  had  already  found 
place  in  the  premisses. 


CHAPTER  III. 


THE   CONVERSION   OF   COMPLEX   PROPOSITIONS. 


309.  If  from  No  A  is  EC,  I  infer  that  No  B  is 
A  C,  what  is  the  nature  of  the  inference  ?  [v.] 

This  inference  is  of  the  nature  of  Conversion,  but  three 
terms  being  involved,  it  is  necessarily  more  complex  than 
those  cases  of  conversion  which  have  been  previously  con- 
sidered.    It  may  be  simply  analysed  as  follows, — 

No  A  is  both  B  and  C, 
therefore,  Nothing  is  at  the  same  time  A,  B,  and  C, 

therefore,  No  B  is  both  A  and  C 
The  reasoning  may  also  be  resolved  into  a  series  of 
ordinary  conversions : — 

No  A  is  BC, 
therefore  (by  conversion),  No  ^Cis  A, 
ue.,  within  the  sphere  of  C,  No  B  is  A, 
therefore  (by  conversion),  within  the  sphere  of  C,  No  A  is  B, 

i.e.,  No  AC  is  B, 
therefore  (by  conversion),  No  B  is  AC. 


312 


COMPLEX   INFERENCES. 


[part  IV. 

Or,  it  may  be  treated  thus, — 

No  A  is  BC, 

therefore  a  fortiori,  No  ^Cis  BC\ 

therefore,  No  AC  \^  B,  (for  if  any  AC  were  B,  it  would 

necessarily  be  BC)\ 
therefore  (by  conversion),  No  B  is  AC 

310.  The  application  of  the  term  Conversion  to 
propositions  containing  more  than  two  terms. 

Generalising,  we  may  say  that  we  have  a  process  of  Con- 
version when  from  a  given  proposition  we  infer  a  new  one 
in  which  a  term  that  appeared  in  the  predicate  of  the 
original  proposition  now  appears  in  the  subject,  or  via 
versa. 

If  a  coffiplex  proposition,  (by  which  I  here  mean  a 
proposition  containing  more  than  two  terms),  is  a  universal 
negative,  any  term  may  be  transferred  from  subject  to  pre- 
dicate or  vice  versa  laithout  affecting  the  force  of  the  asser- 
tion. 

We  have  just  shewn  how  from 

No  A  is  BC, 
we  may  obtain  by  conversion 

No  j^is^C. 
Similarly,  we  may  infer 

No  C  is  AB, 
No  AB  is  C, 
No^Cis^, 

No  BC  is  A. 

The  proposition  might  also  be  written, — 

There  is  no  ABC, 
or,  Nothing  is  at  the  same  time  A,  B  and  C. 

m 

*  Cf.  sections  293,  294. 


CHAP,  ni.]  COMPLEX   INFERENCES.  3i3 

'^The  application  of  the  process  of  Conversion  to  affirma- 
tive propositions  is  of  less  importance;  since  the  converse  of 
an  affirmative  proposition  whether  simple  or  complex  is 
always  particular.  Particular  propositions  are  not  m  them- 
selves of  great  value ;  and,  as  shewn  in  Part  n.,  chapter  v.n 
they  may  involve  us  in  troublesome  questions  with  regard 

to  "  existence." 

In  dealing  with  complex  propositions,  it  is  especially 
desirable,  or  even  essential,  to  keep  clear  of  the  implication  ^^ 
of  the  existence  of  the  subject  of  the  proposition.    I  proceed      ^ 
always  on  the  hypothesis  that  a  universal  proposition  in  no    . 
case  does  more  than  negative  the  existence  of  certain  conv  , 

binations.  Thus,  No  A  is  BCD  negatives  ABCD;  All 
AB  is  CD  negatives  ABc  and  ABd,  (as  usual  denoting 
not-C  by  c  and  not-Z>  by  ti). 

It  is  worth  while  pointing  out  that  from  All  ^  is  ^C  we 
may  obtain  by  conversion  Some  ^  is  ^C,  and  Some  C 
is  AB  ■  but  in  complicated  inferences  we  shall  hardly  ever 
have  occasion  to  convert  affirmative  propositions  in  this 
way  We  shall  find  however  that  to  counterbalance  this, 
the  process  of  contraposition  is  particularly  valuable  in  its 


U 


application  to  complex  universal  affirmative  propositions. 


311.  Shew  clearly  that  if  No  De  is  ABc,  then 
No  ABcD  is  . ;  if  No  r  is  Bdk,  then  No  ^^J^  .^ ;  if 
No  AbDF  is  AT.  then  No  AbcDE  is  FK;  if  ABCis  EF, 
then  ABCG  is  BE;  if  No  AbDE  is  bCE,  then  No 
CDEF  is  AbH. 


CHAPTER  IV. 


THE  OBVERSION  OF   COMPLEX  PROPOSITIONS. 

312.     The  Obversion  of  Propositions  containing 
more  than  two  terms. 

The  doctrine  of  Obversion  is  immediately  applicable  to 
Complex  Propositions ;  and  we  require  no  modification  of 
our  former  definition  of  Obversion.  From  any  given  pro- 
position we  may  infer  a  new  one  by  changing  its  quality  and 
taking  as  a  new  predicate  the  contradictory  of  the  original 
predicate.  The  proposition  thus  obtained  is  called  the 
obverse  of  the  original  proposition. 

The  only  difficulty  connected  with  the  obversion  of 
complex  propositions  consists  in  finding  the  contradictory 
of  a  complex  term.  We  have,  however,  in  section  288,  given 
a  simple  rule  for  finding  the  contradictory  of  any  complex 
term  \~For  each  simple  term  involved,  stibstitute  its  contra- 
dictory ;  write  2ind/or  or,  and  or  for  and. 

Applying  this  rule  to  ''AB  or  ^/;,"  we  have  "(^  or  b) 
and  {A  or  ^),"  i.e.,  ^'Aa  or  Ab  or  aB  or  Bb'';  but  since 
Aa  and  Bb  involve  self-contradiction,  they  may,  as  shewn 
in  section  293,  be  omitted.  The  obverse,  therefore,  of  '*  All 
X is  AB  or  ab "  is  "No  X is  Ab  or  aB,'' 


CHAP.  IV.]  COMPLEX   INFERENCES.  S^S 

313.  Find  the  obverse  of  each  of  the  following 
propositions : — 

(i)  A  isBQ 

(2)  A  is  BC  or  BE, 

(3)  No  A  is  BcE  or  BCF, 

(4)  No  A  is  B  or  bcBEf  or  dcdEF, 

(j)    ''A  is  BC  gives  at  once  "  J^o  A  is  b  or  r." 

(2)  ''A  is  BC  or  jDE''  gives  "No  A  is  {b  or  c)  and  at  the 
same  time  {d  or  ^)."  As  already  pointed  out,  it  may  be 
necessary  to  use  brackets  in  this  way  to  avoid  ambiguity. 
Without  brackets,  however,  and  avoiding  all  chance  of 
ambiguity,  we  may  write  the  above,— "No  A  is  bd  or  be  or 
cd  or  ce,''  The  student  should  make  it  very  clear  to  him- 
self that  these  two  forms  are  really  equivalent. 

(3)  "  No  A  is  BcE  or  BCEJ'  Here  by  the  application 
of  the  general  rule  we  have  as  the  contradictory  of  the 
predicate, — ''(b  or  C  or  e)  and  at  the  same  time  (b  or 
c  or  /)."  What  is  "both  b  and  b''  is  of  course  "/^," 
and  we  have  no  more  information  about  a  thing  if  we 
are  told  that  it  is  "both  b  and  b"  than  if  we  are  told 
that  it  is  simply  "^";  it  has  also  been  already  pointed  out 
that  such  a  term  as  Cc  must  represent  what  is  non-existent, 
and  therefore  when  it  is  given  as  one  among  several 
alternatives  it  may  be  neglected ;  again  as  shewn  in  section 
293  such  an  expression  as  "  /^  or  be  or  bf  or  Cf"  may 
be  simplified  to  '' b  or  C/:'  Remembering  these  three 
points,  we  find  that  "  (b  or  C  or  e)  and(^  or  c  or/)"  maybe 
written  "/^  or  Cf  or  ce  or  efr  For  the  obverse  of  the  given 
proposition,  we  have,  therefore,  ^'A  is  b  or  Cf  or  ce  or  ef^ 

(4)  "  No  y^  is  ^  or  bcDEf  or  bcdEF:'  The  obverse 
is,— «^  is  b  and  {B  or  C  or  d  or  e  or  F)  and  {B  or  C  or 
Dore  or/)";  i.e.,  "^  is  bC  or  bDF or  be  or  bdf:' 


3i6  COMPLEX   INFERENCES.  [part  iv. 

314.  Find  the  obverse  of  each  of  the  following^ 
propositions : — 

(1)  Nothing  is  X,  YorZ; 

(2)  X  is  Ad  or  aC; 

(3)  W  is  XZ  or  Kr  or  VZ  or  Xy  or  xZ ; 

(4)  A  5  is  CDE/  or  C^  or  cD/  or  r^^" ; 

(5)  No  Deis  ABC  or  Adc; 

(6)  No  ^  is  Cd  or  rZ>  or  ^.r^/. 

315.  No  citizen  is  at  once  a  voter,  a  house- 
holder and  a  lodger ;  nor  is  there  any  citizen  who  is 
neither  of  the  three. 

Every  citizen  is  either  a  voter  but  not  a  house- 
holder, or  a  householder  and  not  a  lodger,  or  a  lodger 
without  a  vote. 

Are  these  statements  precisely  equivalent  ?    [v.] 
It  may  be  shewn  that  each  of  these  statements  is  the 

logical  obverse  of  the  other.     They  are  therefore  precisely 

equivalent. 

Let  F=  voter, 

11=  householder, 
Z  =  lodger, 

The  first  of  the  given  statements  is 

No  Citizen  is  FIf£  or  v/i/; 

therefore  (by  obversion),  Every  citizen  is  either  z;  or  ^  or 
/  and  is  also  either  V  or  If  or  Z; 

therefore    (combining  these  possibilities),  Every  citizen   is 
either  Hv  or  Zz;  or  P7i  or  L/i  or  F/  or  If/. 

But  (by  the  law  of  Excluded  Middle),  Ifv  is  either  HZz^ 
or  If/v; 


z;  =  not  voter; 
//  =  not  householder; 
/=  not  lodger. 


CHAP.  IV.]  COMPLEX   INFERENCES.  31? 

therefore,  Hv  is  Li'  or  If/. 

Similarly,  L/i  is  F/i  or  Lv; 

and  F/  is  If/  or  F/i, 

Therefore,         Every  citizen  is  F/i  or  If/ or  Lv, 
which  is  the  second  of  the  given  statements. 

Again,  starting  from  this  second  statement,  it  follows 
(by  obversion)  that  No  citizen  is  at  the  same  time  v  or  ZT, 
h  or  Z,  /  or  F; 
therefore,  No  citizen  is  v/i  or  vL  or  HL^  and  at  the  same 

time  /  or  F\ 

therefore.  No  citizen  is  vh/  or  FHL, 

which  brings  us  back  to  the  first  of  the  given  statements. 

316.     Shew  that  the  two   following   propositions 
are  equivalent : — 

No  X  is  y^  or  BC  or  BD  or  DE, 
X  is  aBcd  or  abDe  or  abd. 


CHAPTER  V. 


THE    CONTRAPOSITION    OF    COMPLEX    PROPOSITIONS. 


317.  The  application  of  the  term  Contraposition 
to  propositions  containing  more  than  two  terms. 

According  to  our  original  definition,  we  contraposit  a 
proposition  when  we  infer  from  it  a  new  proposition  which 
has  the  contradictory  of  the  old  predicate  for  its  subject 
and  the  old  subject  for  its  predicate. 

Thus,  "No  not-i?  is  A''  is  the  contrapositive  of  "All^ 
is^";  "All  not-^  is  not-^"  is  its  obverted  contrapositive. 
Similarly,  the  contrapositive  of  "^  \s  ^  or  C"  would  be 
"No  Ifc  is  A",  the  obverted  contrapositive  "^^  is  a".  The 
contrapositive  of  "^  is  BC'^  would  be  *'No  If  or  c  is  A.'* 
It  will  be  observed,  therefore,  that  the  old  rule  for  obtain- 
ing the  contrapositive  still  applies,  namely, — first  obvert  the 
given  proposition,  and  then  convert  it. 

The  contrapositive  of  a  negative  proposition  is  as  before 
particular,  and  may  be  practically  neglected. 

The  following  simple  rule  may  then  be  given  for  ob- 
taining the  obverted  contrapositive  of  a  universal  affirmative 
proposition  : —  Take  as  a  neiv  subject  the  contradictory  of  the 
old  predicate^  and  as  a  new  predicate  the  contradictory  of  the 


CHAP,  v.]  COMPLEX   INFERENCES.  319 

old  subject^  the  proposition  still  remaini?ig  affirmative.      For 
example, — 

A  is  BCy  therefore,  whatever  I?,  b  ox  c  is  a, 

A  is  B  or  C,  therefore,  be  is  a. 

A  is  BC  ox  Ey  therefore,  whatever  is  be  or  ce  is  a. 

So  far  I  have  been  discussing  what  may  be  called  the 
full  contrapositive  of  a  complex  proposition ;  and  starting 
with  a  universal  affirmative  we  can  pass  back  from  such  a 
contrapositive  to  the  original  proposition.  In  other  words, 
any  universal  affirmative  proposition  and  its  full  contra- 
positive are  equivalent  propositions. 

In  relation  to  complex  propositions,  however,  we  shall 
find  it  convenient  to  give  the  term  Contraposition  an  ex- 
tended meaning.  We  may  say  that  we  have  a  process  of 
Contraposition  when  fro  jn  a  given  proposition  we  infer  a  new 
one  in  which  the  contradictory  of  a  term  that  appeared  in  the 
predicate  of  the  original  proposition  now  appears  in  the  subject^ 
or  the  cofitradictory  of  a  term  that  appeared  in  the  subject  of 
the  original  proposition  now  appears  in  the  predicate. 

We  may  distinguish  four  operations  which  will  be  in- 
cluded under  this  definition : 

(i)  The  operation  of  obtaining  \hQ  full  contrapositive 
of  a  given  proposition,  as  above  described  and  defined. 

(2)  From  '^A  is  BC  ox  E'\  we  may  infer  '*  whatever  is 
be  or  ce  is  a''\  but  in  a  given  appHcation  it  may  be  sufficient 
for  us  to  know  that  "  ^^  is  ^  ",  and  although  this  is  not  the 
full  contrapositive  of  the  original  proposition,  we  may 
regard  it  as  immediately  obtained  from  the  original  pro- 
position by  a  process  of  contraposition. 

With  reference  to  this  case,  the  following  general  rule 
may  be  given, — If  one  or  other  of  a  series  of  alternatives  is 


320 


COMPLEX   INFERENCES. 


[part  IV. 


CHAP,  v.] 


COMPLEX   INFERENCES. 


321 


predicated  of  a  subject,  the  contradictory  of  this  subject  may  be 
predicated  of  any  tenn  that  is  incompatible  with  all  these  alter- 
natives. Thus,  if  "  A  is  PqR  ox  pRS'\  we  may  infer  that  ''ps 
is  ^";  since /J  is  neither  PqR  nor  pRS,  and  therefore  what 
is  ps  cannot  be  A.  But  we  have  not  here  the  full  contra- 
positive  of  the  given  proposition,  and  we  could  not  pass 
back  from  '' ps  is  ^"  to  ''A  is  PqR  or  pRS'\  but  only  to 

''AisPox  sr 

(3)  From  the  proposition  *'A  isB  or  C",  it  follows  that 
if  A  is  not  B,  it  is  C;  but  this  is  expressed  by  the  proposition 
^'Ab  is  C"j  and  the  contradictory  of  a  term  that  originally 
appeared  in  the  predicate  now  appears  in  the  subject, — 
i.e.y  according  to  the  above  definition  we  have  a  process  of 
contraposition.  This  process  might  also  be  described  as  the 
omission  of  one  or  more  of  a  series  of  alternatives  in  the  predi- 
cate by  a  further  particiilarisation  of  the  subject. 

With  reference  to  this  case,  the  following  general  rule 
may  be  given, — If  any  term  X  is  combined  ivith  every  al- 
ternative in  the  subject  of  a  proposition^  every  alternative  in 
the  predicate  which  contains  the  contradictory  of  this  term  fnay 
he  07nitted\     Thus,  from 

Whatever  is  ^  or  ^  is  C  or  DX  or  Ex, 
we  may  obviously  infer 

Whatever  is  AX  or  BX  is  C  or  D, 

1  This  may  sometimes  result  in  the  disappearance  of  all  the  alterna- 
tives; and  the  meaning  of  such  result  is  that  we  now  have  a  non- 
existent subject. 

Thus,  given 

F  is  ABCD  or  Abed  or  aBCd, 

if  we  particularise  the  subject  by  making  it  PdC,  we  find  that  all  the 
alternatives  in  the  predicate  disappear.  The  interpretation  is  that  the 
class  FbC  is  non-existent,  i.e.^  No  P  is  dC ;  a  conclusion  which  of 
course  might  also  have  been  obtained  directly  from  the  given  proposi- 
tion. 


(4)  The  last  operation  to  which  reference  is  made 
above  is  the  reverse  of  that  which  we  have  just  discussed. 
From  the  proposition  ^^AB  is  C\  we  may  infer  "^  is  ^  or 
C".  This  may  be  described  as  a  generalisation  of  the  subject 
by  the  addition  of  one  or  more  alternatives  in  the  predicate. 
But  it  is  also  clear  that  it  comes  under  the  extended  mean- 
ing that  we  have  given  to  the  term  Contraposition. 

To  meet  this  case,  the  following  general  rule  may  be  /y  olLaJ^ 
given, — Any  term  that  appears  An  the  subject  of  a  proposition  y 

may  be  dropped  therefrom^  if  its  contradictory  is  at  the  same 
time  added  as  afi  additional  alternative  in  the  predicate. 

The  following  may  be  taken  as  typical  examples  of  all 
the  operations  that  we  now  include  under  contraposition: — 

AB  is  CD  or  de\ 

therefore,  yfr^/,  anything  that  is  either  cD  or  cE  or  dE  is  a 
or  ^,  (the  /////  contrapositive,  obverted,  according  to  our 
original  definition); 
secondly y  cE  \%  a  ot  b\ 

thirdly,  ABD  is  C; 

fourthly y  A  is  b  or  CD  or  de. 

Combinations  of  the  third  and  fourth  operations  give 

AcD  is  b ; 

Ad  is  ^  or  ^; 
&c. 
The  first  of  the  above  being  called  the  Bull  Contra- 
positive  of  the  given  proposition,  the  remaining  inferences 
may  be  called  Partial  Contrapositives,  according  to  our 
extended  definition  of  contraposition.  In  each  case,  some 
term  disappears  from  the  subject  or  from  the  predicate  of 
the  original  proposition,  and  is  replaced  by  its  contradictory 
in  the  predicate  or  the  subject  accordingly.  Only  in  the 
full  contrapositive,  however,  is  every  term  thus  transposed. 


K.  L. 


21 


322 


COMPLEX  INFERENCES. 


[part  IV. 


I  do  not  think  that  any  confusion  need  result  from  the 
nomenclature  now  proposed,  since  the  extended  use  of  the 
term  Contraposition  can  be  applied  only  to  complex  propo- 
sitions. There  is  still  only  one  kind  of  Contraposition 
possible  in  the  case  of  the  categorical  proposition  containing 
but  two  terms. 

The  great  importance  of  Contraposition  as  we  are  now 
dealing  with  it  in  connection  with  complex  propositions  is 
that  by  its  means,  given  a  universal  affir7tiative  proposition 
of  any  complexity,  ive  may  obtain  separate  i7ifor7nation  with 
regard  to  any  term  that  appears  in  the  subject,  or  ivith  regard 
to  the  contradictory  of  any  term  that  appears  in  the  predicate, 
or  with  regard  to  any  cotnbination  of  these  terms.  Thus, 
given  "XF  is  P  or  Qr'\  by  the  process  described  as  the 
generalisation  of  the  subject,  we  have 

X\%  y  ox  P  ox  Qr, 

The  particularisation  of  the  subject  gives 

XYp  is  Qr, 
XYq  is  P, 
&c. ; 
and  by  the  combination  of  these  processes,  we  have 

XpY^y  ox  Qr\ 
S:c. 

Again,  the  full  contrapositive  of  the  original  proposition 
is 

Whatever  ispq  ox  pR  is  x  oxy, 

from  which  we  have 

pis  X  ox  y  ox  Qr^ 

qis  X  ox  y  ox  /*, 

&c. 


CHAP,  v.]  COMPLEX   INFERENCES.  323 

318.  Given  "All  D  that  is  either  B  ox  C  is  ^," 
shew  that  "  Everything  that  is  not-^  is  either  not-5 
and  not-C  or  else  it  is  not-A"  [De  Morgan.] 

This  example  and  the  five  following  examples  are  adapt- 
ed from  De  Morgan,  Syllabus,  p.  42.  They  are  also  given 
by  Jevons,  Studies,  p.  241,  in  connection  with  his  Equational 
Logic.     They  are  all  simple  exercises  in  Contraposition. 

We  have,  What  is  either  BD  or  CD  is  A, 
therefore,  a  is  {p  or  d)  and  (c  or  ^), 
therefore,  a  is  be  or  d. 

319.  Given  "  All  A  is  either  BC  or  BD;'  shew 
that  ''All  that  is  not-i?  and  not- (7  is  not-^"  and  "All 
that  is  not-Z^  is  not- A!'  [De  Morgan.] 

320.  If  A  is  either  BC  or  D,  and  if  whatever  is 
BC  ox  D  is  A,  shew  that  whatever  is  not-^  is  not-D 
and  also  either  not-^  ornot-C  and  whatever  is  not-D 
and  at  the  same  time  not-^  or  not-^'  is  r\ot-A, 

[De  Morgan.] 

321.  If  whatever  is  B  or  CD  or  CE  \s  A,  what 
do  we  know  about  not-^  .?  [De  Morgan.] 

322.  If  whatever  is  either  B  ox  C  and  at  the 
same  time  either  D  ox  E  \s  A,  what  do  we  know 
about  not-^  >  [De  Morgan.] 

323.  If  that  which  is  A  or  BC  and  is  also  D  or 
EF  is  X,  what  is  all  that  we  know  about  not-X } 

[De  Morgan.] 

What  is  {A  or  BC)  and  {D  or  EF)  is  X; 

therefore  (by  contraposition),  x  is  ab  or  ac  or  de  or  df 

21 — 2 


324 


COMPLEX   INFERENCES. 


[part  IV. 


[There  is  apparently  a  misprint  in  Jevons's  transcription 
of  this  example  {Studies,  p.  241).  He  uses  difterent  letters, 
but  his  implied  solution  is  *'  x  is  either  ab  or  c  and  it  is  also 
either  de  or/".     I  cannot  see  how  this  is  obtainable.] 

324.  To  say  that  whatever  is  devoid  of  the  pro- 
perties of  A  must  have  those  either  of  B  or  of  D,  or 
else  be  devoid  of  those  of  Cy  is  the  same  as  to  say- 
that  what  is  devoid  of  the  properties  of  B  and  Dy 
but  possesses  those  of  Cy  must  have  A.     Prove  this. 

[Jevons,  Studies^  p.  239.] 

325.  Shew  that  '' C  is  Ab  or  aB''  is  equivalent 
to  the  two  propositions  '^AB  is  r"  and  ^^ab  is  c", 

[Jevons,  StudicSy  p.  239.] 

326.  Prove  the  equivalence  of  the  following  as- 
sertions : — 

(i)     Every  gem  is  either  rich  or  rare. 

(2)  Every  gem  which  is  not  rich  is  rare. 

(3)  Every  gem  which  is  not  rare  is  rich. 

(4)  Everything  which  is  neither  rich  nor  rare  is 
not  a  gem.  [Jevons,  Studies ,  p.  229.] 

327.  If  that  which  is  devoid  of  heat  and  devoid 
of  visible  motion  is  devoid  of  energy,  it  follows  that 
what  is  devoid  of  visible  motion  but  possesses  energy 
cannot  be  devoid  of  heat.    [Jevons,  Studies,  p.  199.] 

328.  If  the  relations  A  and  B  combine  into  C, 
it  is  clear  that  A  without  C  following  means  that 
there  is  not  By  and  that  B  without  C  following  means 
that  there  is  not  A,  [De  Morgan.] 


CHAP,  v.]  COMPLEX   INFERENCES.  325 

329.  Any  one  who  wishes  to  test  himself  and 
his  friends  upon  the  question  whether  analysis  of  the 
forms  of  enunciation  would  be  useful  or  not,  may  try 
himself  and  them  on  the  following  question : — 

Is  either  of  the  following  propositions  true,  and 
if  either,  which  .'* 

(i)  All  Englishmen  who  do  not  take  snuff  are 
to   be   found    among    Europeans    who  do   not  use 

tobacco. 

(2)  All  Englishmen  who  do  not  use  tobacco  are 
to  be  found  among  Europeans  who  do  not  take 
snuff. 

Required  immediate  answer  and  demonstration. 

[De  Morgan.] 

330.  Is  the  student  of  logic,  generally  speaking, 
prepared  rapidly  to  analyse  the  two  following  pro- 
positions, and  to  say  whether  or  no  they  must  be 
identical,  if  the  identity  of  synonyms  be  granted  > 

(i)  The  suspicion  of  a  nation  is  easily  excited, 
as  well  against  its  more  civilised  as  against  its  more 
warlike  neighbours,  and  such  suspicion  is  with  diffi- 
culty removed. 

(2)  When  we  see  a  nation  either  backward  to 
suspect  its  neighbour,  or  apt  to  be  satisfied  by  ex- 
planations, we  may  rely  upon  it  that  the  neighbour 
is  neither  the  more  civilised  nor  the  more  warlike  of 
the  two.  [De  Morgan.] 

331.  Infer  all  that  you  possibly  can  by  way  of 
Contraposition  or  otherwise,  from  the  assertion,  All 
A  that  is  neither  B  nor  C  is  X.  [R.] 


326  COMPLEX  INFERENCES.  [part  IV. 

The  given  proposition  may  be  written  Abe  is  X-,  and 
taking  it  as  it  stands,  the  converse  is  Some  X  is  Abe,  and  the 
contrapositive  (obverted)  x  is  a  ox  B  ox  C.  We  may  also 
get  a  number  of  other  propositions  with  which  we  may 
proceed  in  the  same  way;  e.g.^ — 

Abx  is  Cy 
Acx  is  By  &c. 

Confining  ourselves  however  to  such  universal  proposi- 
tions as  can  be  obtained,  the  problem  may  be  solved 
generally  as  follows : — 

The  given  proposition  may  be  thrown  into  the  form, — 
Nothing  is  at  the  same  time  A,  b,  c  and  x\ 
and  we  see  that  it  is  symmetrical  with  regard  to  the  terms 
A,  b,  c,  X,  We  are  sure  then  that  anything  that  is  true  of 
A  is  true  mutatis  mutandis  of  b,  c  and  .r,  that  anything  that 
is  true  of  Ab  is  true  7ttutatis  viutandis  of  any  pair  of  the 
terms,  and  similarly  for  combinations  three  and  three  to- 
gether. 

We  have  at  once  the  four  symmetrical  propositions,— 
A\%Bqx  CoxX\    (i) 
^  is  ^  or  Cor  X\    (2) 
c  \?>  a  ox  B  ox  X\     (3) 
^  is  flj  or  j5  or  (7.      (4) 

Then  from  (i)  we  have 

AbisCoxX)  (i) 
and  the  five  corresponding  propositions  are, — 

-^ris^or^;      (ii) 

Ax  is  B  or  C\  (iii) 

be  \?>  a  ox  X )  (iv) 

bx  is  a  or  C\  (v) 

ex  is  a  or  B,  (vi) 


CHAP,  v.]  COMPLEX   INFERENCES.  327 

Again  from  (i), — 

Abe  is  X,  (which  is  the  original  proposition),  (a) 
and  we  have,  similarly, — 

Abx  is  C;  W 

Acx  is  B ;  (y) 

box  is  a,  (S) 

It  should  be  noted  that  the  following  are  pairs  of  contra- 
positives, — 
(i)  (5),  (2)  (7),  (3)  (^),  (4)  (a),  0  (vi),  (ii)  (v),  (iii)  (iv). 

332.  Find  the  /////  contrapositive  of  each  of  the 
following  propositions : — 

A  is  BCDe  or  bcDe  ; 

AB\s  CD  or  cDE  or  de\ 

Whatever  is  AB  or  bC  is  aCd  or  Acd", 

Where  A  is  present  along  with  either  B  or  Cy  D  is 
present  and  C  absent  or  D  and  E  are  both  absent ; 
Whatever  is  ABC  or  abc  is  DEF  or  def. 

333.  Compare  the  logical  force  of  the  following 
propositions : — 

(i)     All  voters  who  are  not  lodgers  are   house- 
holders who  pay  rates ; 

(2)  No  one  who  is  not  a  lodger  and  who  does 
not  pay  rates  is  a  voter ; 

(3)  A  voter  who  is  a  householder  is  not  a  lodger; 

(4)  A  householder  who  does  not  pay  rates  is  not 
a  voter ; 

(5)  All  who  pay  rates  or  are  householders  are 
voters ; 


328  COMPLEX   INFERENCES.  [part  iv. 

^  (6)  Anyone  who  is  not  a  householder  or  who 
being  a  householder  does  not  pay  rates  is  either  not 
a  voter  or  else  he  is  a  loderer : 

(7)  All  who  have  a  vote  pay  rates ; 

(8)  Anyone  who  has  no  vote  is  either  not  a  rate- 
payer or  not  a  householder. 

334.  If  A  is  either  B  or  C,  shew  that  what  is  not 
B  is  either  C  or  not  A. 

335.  What  is  the  difference  between  the  assertion 
that  A  is  BC  and  the  pair  of  assertions  that  d  is  a, 
and  c  is  d?  [Jevons,  Studies,  p.  239.] 

336.  \{  A  unless  it  is  B  is  either  CD  or  EF,  shew 
that  not-C  is  either  not-^  or  B  or  EF, 

337.  Establish  the  followinor, — 

(i)  Where  B  is  absent,  either  A  and  C  are  both 
present  or  A  and  D  are  both  absent;  therefore,  where 
C  is  absent,  either  B  is  present  or  D  is  absent. 

(ii)  Where  A  is  present  and  also  either  B  or  E, 
either  C  is  present  and  D  absent  or  C  is  absent  and  D 
present ;  therefore,  where  C  and  D  are  either  both 
present  or  both  absent,  either  A  is  absent  or  B  and 
E  are  both  absent. 

(iii)  Where  A  is  present,  either  B  and  C  are  both 
present  or  C  is  present  D  being  absent  or  C  is  present 
F  being  absent  or  H  is  present ;  therefore,  where  C  is 
absent,  A  cannot  be  present  H  being  absent. 

338.  Among  plane  figures  the  circle  is  the  only 
curve  of  equal  curvature.     Shew  that  this  is  the  same 


■u- 


CHAP,  v.]  COMPLEX   INFERENCES.  329 

as  to  assert  that  a  plane  figure  must  either  be  a  curve 
of  equal  curvature,  in  which  case  it  is  also  a  circle,  or 
else,  not  a  circle  and  then  not  a  curve  of  equal  curva- 
ture. [Jevons,  Studies,  p.  235.] 

Let /'^  plane  figure, 

C=  circle, 

E  =  curve  of  equal  curvature. 
"Among  plane  figures  the  circle  is  the  only  curve  of 
equal  curvature,"  may  be  expressed  by  BC  is  C£,  and  Be 
is  a.  "A  plane  figure  must  either  be  a  curve  of  equal 
curvature,  in  which  case  it  is  also  a  circle,  or  else,  not  a 
circle  and  then  not  a  curve  of  equal  curvature,"  becomes 
F  is  CE  or  ee.  It  is  immediately  obvious  that  the  two 
statements  are  equivalent. 

339.  ''  Similar  figures  consist  of  all  figures  whose 
corresponding  angles  are  equal  and  whose  sides  are 
proportional."  Give  all  the  propositions  involving 
not  more  terms,  which  can  be  inferred  from  the  above. 
Give  also  one  proposition  equivalent  to  it.  [l.] 

Let  B=  similar  figures, 

<2  =  figures  whose  corresponding  angles  are  equal, 
i?  =  figures  whose  sides  are  proportional. 
The  given  statement  may  be  resolved  into  the  two  pro- 
positions,—  .     ^„ 
^                                   All  F  is  (2^, 

All  QF  is  F, 
From  these,  by  contraposition,  we  may  infer,— 

/  is  ^  or  r; 

QisFFor^n 
^is/i 


330  COMPLEX  INFERENCES.  [part  IV. 

Ji  is  FQ  Qxpq-y 
risp- 
FQ  is  i?; 
FR  is  (2; 
PQ  is  r; 
/i'v'  is  q\ 
Qr  is  J>] 
qR  is/; 
^^z-  is  /. 

Fq  and  TV  represent  non-existent  classes ;  and  we  have  no 
information  with  regard  \o  pq  snidpr. 

This  I  think  affords  a  complete  solution  of  the  first  part 
of  the  question.  We  may  obtain  a  statement  equivalent  to 
the  given  statement,  by  taking  the  full  contrapositives  of 
the  two  propositions  into  which  we  resolved  it  and  then 
combining  them.     Thus, 

Fis  QR,=q  or  rhp, 

QR  is  F,  =/  is  q  or  r; 
and  these  are  combined  in  the  statement  that/  consists  of 
all  things  that  are  q  or  r.     **  Figures  that  are  not  similar 
consist  of  all  figures  whose  corresponding  angles   are   not 
equal  or  whose  sides  are  not  proportional." 

340.  Given  A  is  BQ  what,  if  anything,  do  you 
know  concerning  the  classes  AB,  Ab,  AC,  Ac,  a,  aB, 
ab,  aC,  ac,  B,  BQ  Be,  b,  bC,  be,  C,c> 

A  is  BC, 
therefore,  by  conversion.  Some  ^Cis  A,  (i) 

By  contraposition,  we  may  obtain  the  two  propositions, 

No  b  is  Ay  1 2) 

No  c  is  A,  u\ 


)\ 


(4) 


CHAP,  v.]  COMPLEX   INFERENCES.  33^ 

Then  by  once  more  obverting  and  converting, 

Some  a  is  b^ 
Some  a  is 

We  cannot  combine  these  into  "  Some  a  is  he''  since  we 
do  not  know  that  the  same  a  is  referred  to  in  both  cases. 

The  other  forms  which  can  be  obtained  are  in  reality 
only  weakened  forms  of  one  or  other  of  the  above. 

By  (3)  No  e  is  A, 
i.  e.y  Nothing  is  Ac, 
therefore  {a  fortiori),  No  B  is  Ae, 

Similarly,  by  (2),         No  b  is  A, 

i.  e.,  Nothing  is  Ab, 
therefore  {a fortiori),  No  C  is  Ab. 
Again,  by  obversion  of  the  original  proposition, 

No  A  is  e, 
therefore  {a  fortiori),  No  AB  is  c. 

Similarly,  No  ^  C  is  b. 

Also  from  (2),  No  /^C  is  A, 

and  No  be  is  A, 

Similarly,  from  (3),   No  Be  is  A. 

We  cannot  obtain  any  information  with  regard  to  the 
remaining  classes  aB,  ab,  aC,  ae. 

[As  already  indicated,  I  should  consider  that  (i)  and  (4) 
involve  assumptions  with  regard  to  "  existence."  Without 
any  such  assumptions,  however,  we  can  obtain  all  the  re- 
maining inferences.  We  may  regard  (4)  as  obtained  by 
inversion  of  the  original  proposition.    Cf.  section  348.] 

341.  Assuming  that  armed  steam-vessels  consist 
of  the  armed  vessels  of  the  Mediterranean  and  the 
steam-vessels  not  of  the  Mediterranean,  inquire  whe- 
ther we  can  thence  infer  the  following  results : — 


(5) 
(6) 

(7) 
(8) 


(9) 
(10) 

(II) 
(12) 

(13) 


333 


COMPLEX   INFERENCES. 


[part  IV. 

(i)  There  are  no  armed  vessels  except  steam- 
vessels  in  the  Mediterranean. 

(2)  All  unarmed  steam-vessels  are  in  the  Medi- 
terranean. 

(3)  All  steam-vessels  not  of  the  Mediterranean 
are  armed. 

(4)  The  vessels  of  the  Mediterranean  consist  of 
all  unarmed  steam-vessels,  any  number  of  armed 
steam-vessels,  and  any  number  of  unarmed  vessels 
without  steam.      [Jevons,  Studies,  p.  23 1 ,  from  Boole.] 

342.  If  AB  is  either  Cd  or  cDl\  and  also  either 
eF  or  //,  and  if  the  same  is  true  of  BH,  what  do  we 
know  of  that  which  is  E  ? 

We  have  given, — 

What  is  AB  or  Blfis  (Cd  or  cBe)  and  (^7^  or  //); 

therefore,  What  is  AB  or  BIf  is  CdcF  01  cDeF  ox  CdH  ox 
cDeH', 

therefore.  What  is  ABE  or  BHE  is  CdH-, 
therefore,  E  is  CdH  ox  b  or  ah, 

343.  If  A  that  is  B  is  either  P  ox  Q  and  also 
either  R  or  S,  and  if  the  same  is  true  of  A  that  is  both 
C  and  D,  what  is  all  that  we  know  about  that  which 
is  neither  P  nor  5  ? 

344.  Given  that  whatever  is  PQ  or  AP  is  bCD 
or  abdE  or  aBCdE  or  Abed,  shew  that,— 

(1)  abP\s,  CD  ox  dE  or  q; 

(2)  DP\^bCoxaq', 

(3)  Whatever  is  B  or  Cd  or  cD  is  aox  p\ 

(4)  B  is  C  ox  p  or  aq\ 


CHAP,  v.]  COMPLEX  INFERENCES.  333 

(5)  Cd\saoxp\ 

(6)  AB\sp\ 

(7)  If  ^^  is  ^  or  ^  it  is /  or  <7 ; 

(8)  If  BP  is  e  ox  D  ox  e  it  is  aq. 

345.  Given  A  is  BC  or  BBE  ox  BDF,  infer  de- 
scriptions of  the  following  terms  Ace,  Acf,  ABcD, 

[Jevons,  Studies,  pp.  237,  238.] 

In  accordance  with  rules  already  laid  down,  we  have 

immediately, — 

Ace  is  BDF\ 

AcfisBDE) 

ABcD  isEox  F 

346.  Given  d  is  CDe  ox  Acd  or  adcF  or  acdEF, 
infer  descriptions  oi  A,  bf,  CD,  cD,  de. 

347.  Given  that  PQr  is  ABc  or  abD  or  aCDE  or 
BCdeF  or  bCdf  ox  CDEF  ox  def,  what  is  all  that  you 
know  concerning  the  classes, — 

A,a,B,b,  C,c,D,d,E,e,F,f} 

348.  The  Inversion  of  Complex  Propositions. 
We  might  define  Inversion  in  connection  with  complex 

propositions  as  a  process  by  which  from  a  given  proposition 
we  infer  a  new  one  in  which  some  term  in  the  subject 
is  replaced  by  its  contradictory.  I  have  not,  however, 
thought  it  worth  while  to  give  any  detailed  discussion  of 
inversion  here,  because  this  process  always  results  in  par- 
ticular propositions ;  these  are  of  small  importance  at  the 
best,  and  they  involve  assumptions  concerning  the  existence 
of  their  subjects,  which  are  so  inconvenient  when  these 
subjects  are  very  complex,  that  they  are  best  neglected 
altogether,  unless  a  very  special  treatment  is  accorded  to 
them. 


334 


COMPLEX   INFERENCES. 


[part  IV. 


349.     Summary  of  the  results  obtainable  by  Con- 
version, Obversion,  and  Contraposition. 

(i)    By  Conversion  of  a  universal  negative  we  can  obtain 
separate  information  with  regard  to  any  term  that  appears 
either  in  the  subject  or  in  the  predicate,  or  with  regard 
to  any  combination  of  these  terms. 
For  example,  No  AB  is  CD ; 

therefore,  No  A  is  BCD, 
No  C  is  ABB, 
l>lo  BB>  is  AC. 

(2)     By  Obversion  we  can  change  any  proposition  from 
the  affirmative  to  the  negative  form,  or  vice  versa. 

For  example,  AB  is  CD  or  EF)  therefore.  No  AB  is  ce 
or  cfoxde  or  df. 

No  Pis  QRi 
therefore,  Z'  is  ^  or  r, 

(3)  By  Contraposition  of  a  universal  affirmative  we  can 
obtain  information  with  regard  to  any  term  that  appears  in 
the  subject,  or  with  regard  to  the  contradictory  of  any  term 
that  appears  in  the  predicate,  or  with  regard  to  any  combi- 
nation of  these  terms. 

For  example,       AB  is  CD  or  EF-, 

therefore,  ^  is  <^  or  CD  or  EF, 

CIS  a  or  b  ox  EF^ 

Be  is  a  or  CD, 

ce  is  a  or  b, 

Adf  is  b, 

&c. 


CHAPTER  VI. 


THE    COMBINATION    OF    COMPLEX   PROPOSITIONS. 


350.  The  Combination  of  Universal  Affirmative 
Propositions  the  subjects  of  which  do  not  contain 
Contradictories. 

ZisP,  orP,  or  ...  or  P^, 
Fis  <2i  or  Q^ox  ...  or  (2„, 
may  be  taken  as  types  of  two  such  propositions. 

By  combining  them  we  have 

A'Kis  (P,  or  7^2  or  ...  or  PX  and  also 

(CiOr  Q^ox  ...  or  (2,.); 

i.e.,  XY\s  F^Q,  or  F^Q^  or  ...  or  F^Q, 

oxF^Q,oxF^Q,ox  ...oxF^Q, 


or 


or  F^Q,  or  F^Q^  or  ...  or  F^Q„, 

If  the  subject  of  both  the  original  propositions  had  been 
Xj  then  of  course  we  should  have 

X  is  F^Q,  ox  F,Q,  ox &c. 

In  this  case,  the  new  proposition  is  equivalent  to  the  two 
propositions  with  which  we  started,  i.e.,  we  could  pass  back 


336 


COMPLEX   INFERENCES. 


[part  IV. 


from  It  to  them.  But  when  the  subjects  of  the  original 
propositions  are  not  the  same,  the  new  proposition  is  not 
equivalent  to  them. 

In  combining  propositions,  the  student  should  never 
lose  an  opportunity  of  simplifying  his  results ;  and  such 
opportunities  will  be  found  to  be  of  continual  recurrence. 

The  following  are  examples  : 

(i)        A  is  Cor  D, 
B\s  cE\ 

therefore,  AB  is  cDE, 
since  the  combination  of  C  and  cE  is  self-destructive. 

(2)  y^  is  ^  or  C, 
A  \s  c  ox  D; 

therefore,  A  is  Be  or  BD  or  CD, 

(3)  X  is  AB  or  bee, 
Y'lsaBCox  DE] 

therefore,  XY'is  ABDE-, 
for  again  it  will  be  found  that  all  the  other  combinations  in 
the  predicate  contain  contradictories. 

(4)  -Y  is  ^  or  Be  or  Z>, 
Kis  aB  or  Be  or  Cd\ 

therefore,  ^F  is  ^^  or  aBD  or  AC  it 
The  alternatives  in  full  are 

AaB  or  ABe  or  A  Cd  or  aBe  or  Be  or  BeCd  or  aBD 

or  BeD  or  CdD. 

But  AaB,  BeCdy  CdD  represent  non-existent  classes  and 
may  therefore  be  omitted.  ABe,  aBe,  BeD  are  merely 
partial  repetitions  oi  Be,  and  therefore  they  too  may  be 
omitted.     Compare  section  293. 

After   a  very  little   practice,  the    student  will   find  it 
unnecessary  to  write  out  the  alternatives  in  full. 


CHAP.  VI.]         COMPLEX  INFERENCES.  337 

(5)     X  IS  A  or  bd  or  eE^ 
Vis  AC  or  aBe  or  d', 
therefore,  XY \s  AC  or  bd  or  Ad  or  cdE, 

351.     The    Combination   of   Universal   Negative 
Propositions  the   subjects  of  which  do  not  contain 

Contradictories. 

^oX\sP, 
No  Fis  Q, 

may  be  taken  as  types  of  two  such  proposidons. 

By  combining  them  we  have  simply, — 

No  XY  is  either  P  or  Q. 

The  following  are  examples  : 

(i)     No  A  is  cd. 

No  ^  is  C  or  e\ 

therefore,  No  AB  is  either  C  or  ^  or  ed, 

(2)  No  A  is  be. 
No  A  is  Cd\ 

therefore,  No  A  is  be  or  Cd. 

(3)  No  X  is  either  aB  or  aC  or  bC  or  aE  or  bE, 
No  Y  is  either  Ad  or  Ae  or  bd  or  be  or  cd  or  ee ; 

therefore.  No  XY  is  either  aB  or  aC  or  bC  or  aE  or  bE 
or  Ad  or  Ae  or  bd  or  be  or  cd  or  <r^ ; 
therefore,  No  X Fis  either  a  or  b  or  d  or  e\ 

(4)  No  X  is  ^^^  or  aCd, 

No  FisZ^tTor^Z^or^Ci^; 

therefore.  No  XY  is  ^7<^^  or  aCd  or  /^^  or  ^Z>  or  ACD. 

(5)  No  X  is  aBC  or  ^CZ>  or  aBe  or  fltZ^^-, 

No  Fis  AeD  or  ^/^Z>  or  ^rZ>  or  aDE  or  ^Z>^ ; 
therefore.  No  XY\s  aBC  or  aD  or  ^Z>  or  ^^^'. 


1  Cf.  section  302. 
K.  L. 


2  Cf.  section  303. 


22 


33^  COxMPLEX   INFERENCES.  [part  i v. 

352.  The  Combination  of  Propositions  the  sub- 
jects of  which  contain  Contradictories. 

Such  propositions  cannot  be  directly  combined  in  the 
manner  just  discussed.  If  AB  is  Z>,  and  ^C  is  ^,  we  are 
not  really  given  any  information  by  being  told  that  what  is 
both  A B  and  I? Cis  £>!!:. 

To  avoid  this  difficulty  we  must  by  partial  contra- 
position remove  l^of/i  the  contradictories  into  the  predicates 
of  their  respective  propositions. 

Thus,  the  propositions  ^^AB  is  Z>"and  "/^(7  is  JS"  may 
be  reduced  to  the  forms  "^  is  <^  or  Z) "  and  ^^ Cis  B  or  £" ; 
and  we  then  have  by  combining  them,  *'AC  is  l?E  or  BD 
or  Z>i^". 

Starting  with  such  a  pair  of  propositions  as  the  above, 
it  is  requisite  to  take  do//i  the  contradictories  into  the 
predicates,  or  we  shall  still  be  left  with  a  merely  identical 
proposition.  For  example,  combining  "^Z?  is  U''  and  "C 
is  B  or  ^",  we  have  ''ABC  is  ^  or  Z>  or  £",  which  ob- 
viously tells  us  nothing. 

If,  however,  the  propositions  can  be  reduced  to  such  a 
form  that  the  subject  terms  so  far  as  they  are  not  contra- 
dictories are  the  same,  the  predicates  also  being  the  same ; 
then  we  may  obtain  a  new  proposition  by  just  omitting  the 
contradictory  terms.  Thus  if  we  have  propositions  of  the 
form  AB  is  C,  Ai?  is  C,  we  may  infer  (since  A  is  AB  or  Al^) 
that  ^  is  C 

The  same  result  is  also  obtainable  by  means  of  the  rule 
previously  given, — 

AB  is  C,  and  Ad  is  C, 

therefore,  -^  is  ^  or  C,  and  ^  is  ^  or  C, 

therefore,  A  is  BC  or  bC  or  C, 

therefore,  A  is  C 


CHAPTER   VII. 


INFERENCES     FROM     COMBINATIONS     OF     COMPLEX 

PROPOSITIONS. 

353.  Problem,— G'w^xv  any  proposition,  and  any 
term  X,  to  discriminate  between  the  cases  in  which 
the  proposition  does,  and  those  in  which  it  does  not, 
afford  information  with  regard  to  this  term. 

We  may  assume  that  the  original  proposition  is  not  an 
identical  proposition. 

If  it  is  negative,  let  it  by  obversion  be  made  affirmative. 

Then,  written  in  its  most  general  form,  it  will  be 
Whatever  is  P,P^ ...ox  Q,Q,...ox  &c.  is  ^,^, . . .  or  B,B^ . . . 

or  &c. 

As  shewn  in  section  291,  this  may  be  resolved  into  the 
independent  propositions  : — 

P^P^ ...  is  A^A.^ ...  or  B^B^ ...  or  &c. ; 
Q^Q^ ...  is  A^A^  ...  or  B^B^  ...  or  &c. ; 
&c.  &c.  &c.; 

in  none  of  which  is  there  any  disjunction  in  the  subject. 

We  may  deal  with  these  propositions  separately,  and  if 
any  one  of  them  affords  information  with  regard  to  X,  then 
of  course  the  original  proposition  does  so. 

We  have  then  to  consider  a  proposition  of  the  form 

P,P^ ...  P„  is  A^A, ...  or  B,B^ ...  or  &c. 

22 — 2 


340  COMPLEX  INFERENCES.  [part  IV. 

From  this  by  contraposition  we  get, — 
Everything  is  A^A^ ...  or  B^B^ ...  or  &c.  or/,  or/,  ...  or/j 
and  hence,  X  is  A^A^ ...  or  B^B^ ...  or  &c.  or/,  or  /^ . . .  or/^. 

We  may  now  strike  out  all  alternatives  in  the  predicate 
which  contain  x. 

If  they  a//  contain  x,  then  the  information  afforded  us 
with  regard  to  X  is  that  it  is  non-existent. 

If  some  alternatives  are  left,  then  the  proposition  will 
afford  information  concerning  X  unless,  when  the  predicate 
has  been  simplified  to  the  fullest  possible  extent,  one  of 
the  alternatives  is  itself  Jf  uncombined  with  any  other  term, 
in  which  case  it  is  clear  that  we  are  left  with  a  merely 
identical  proposition. 

Now  one  of  these  alternatives  will  be  X  in  any  of  the 
following  cases,  and  only  in  these  cases : — 

J^irsf,  If  one  of  the  alternatives  in  the  predicate  of  the 
original  proposition,  when  reduced  to  the  affirmative  form, 
isX 

Secondly,  If  any  set  of  alternatives  in  the  predicate  of 
the  original  proposition,  when  reduced  to  the  affirmative 
form,  constitute  a  development  of  X\  (since  ^'AX  ox  aX" 
is  equivalent  to  X ]  ''ABX  or  AbX  or  aBX  or  abX''  is 
also  equivalent  to  X',  and  so  on). 

Thirdly,  If  one  of  the  alternatives  in  the  predicate  of 
the  original  proposition,  when  reduced  to  the  affirmative 
form,  contains  X  in  combination  solely  with  some  term 
or  terms  appearing  also  in  the  subject ;  since  in  this  case 
such  alternative  is  equivalent  to  X  simply. 

For  example, 

''AB  is  AX  or  Z>"  is  equivalent  to  ''AB  is  X  or  Z>." 


CHAP.  VII.]         COMPLEX  INFERENCES.  34i 

By  contraposition  of  this  proposition  in  Its  original  form 

we  have, — 

Everything  is  AX  or  D  ox  a  or  b, 

but,  (cf  section  293),  ''AX  ox  a''  is  equivalent  to  "Zor  ^." 

Fourthly,  If  one  of  the  terms  originally  contained  in  the 
subject  is  x\  since  in  that  case  we  should  after  contra-* 
position  have  X  as  one  of  the  alternatives  in  the  predicate. 

The  above  may  now  be  summed  up  in  the  propo- 
sition : — 

Any  ?io?t-identical proposition  will  afford  information  7uiih 
regard  to  any  term  X,  unless  {after  it  has  been  brought  to  the 
affirmative  for  7n),  (i)  one  of  the  alternatives  in  the  predicate  is 
Xj  or  (2)  afiy  set  of  alternatives  in  the  predicate  constitute  a 
development  of  X,  or  (3)  any  alternative  in  the  predicate  con- 
taifis  X  in  combination  with  such  ter?ns  only  as  appear  also  in 
every  alternative  in  the  subject,  or  (4)  every  alternative  in  the 
subject  contains  x. 

If,  after  the  proposition  has  been  reduced  to  the  affirma- 
tive form,  the  simplifications  noticed  in  section  293  have 
been  effected,  then  the  criterion  becomes  more  simply, — 

Any  non-identical  proposition  will  afford  information  with 
regard  to  aiiy  term  X,  unless,  {after  it  has  been  brought  to  the 
affirmative  for?n,  and  its  predicate  so  siffipliffed  that  it  contains 
no  superfluous  terms),  one  of  the  alternatives  in  the  predicate  is 
X,  or  et^ery  alternative  in  the  subject  contaifis  x. 

If  instead  of  X  wq  have  a  complex  term  XYZ,  then  no 
part  of  this  term  must  appear  as  an  alternative  in  the  predi- 
cate, and  there  must  be  at  least  one  alternative  in  the 
subject  which  does  not  contain  the  contradictory  of  any 
part  of  this  complex  term  :  i.e.,  no  alternative  in  the  predi- 
cate must  be  X,  V,  or  Z,  and  some  alternative  in  the  subject 
must  contain  neither  x,  y,  nor  z. 

The  above  criterion  is  of  simple  application. 


342 


COMPLEX  INFERENCES. 


[part  IV. 


354.  Say,  by  inspection,  which  of  the  following 
propositions  give  information  concerning  A,  aB,  by 
bCdy  respectively : — 

Ab  IS  bCd  ox  c; 

bd  is  A  or  bC  or  abc ; 

Whatever  is   a  ox  B  \s  c  ox  D  \ 

Whatever  is  ^^  or  be  is  bE  or  cE  ox  e  \ 

X  \s  AX  or  ab  or  Be  or  Cd, 

355.  Problem. — Given  any  number  of  propositions 
involving  any  number  of  terms,  to  determine  what  is 
all  the  information  that  they  jointly  afford  with  regard 
to  any  given  term  or  combination  of  terms  that  they 
contain. 

The  great  majority  of  direct  problems'  involving  com- 
plex propositions  may  be  brought  under  the  above  general 
form.  If  the  student  will  turn  to  Boole,  Jevons,  or  Venn, 
he  will  find  that  it  is  by  them  treated  as  the  central  problem 
of  Symbolic  Logic. 

A  general  method  of  solution  is  as  follows: — 
Let  X  be  the  term   concerning   which   information  is 
desired.     Find   what   information   each   proposition   gives 
separately  with  regard  to  X,  thus  obtaining  a  new  set  of 
propositions  of  the  form 

X  is  P^  or  P^  or  ...  or  /*„. 

This  is  always  possible  by  the  aid  of  the  rules  given  in 

the  preceding  chapters'.     It  should  be  remembered  that  in 

section  353  we  have  discriminated  the  cases  in  which  any 

given  proposition  really  affords  information  with  regard  to  X. 

^  Inverse  problems  are  discussed  in  chapter  Xii. 
2  The  importance  of  these  rules,  especially  of  those  relating  to  Con- 
traposition, is  now  made  more  apparent. 


CHAP.  VII.]         COMPLEX  INFERENCES.  343 

Those  propositions  which  do  not  do  so  may  of  course  be 
altogether  left  out  of  account. 

Next  combine  the  propositions  thus  obtained  in  the 
manner  indicated  in  section  350.  This  will  give  the  desired 
solution. 

The  method  might  be  varied  by  bringing  the  proposi- 
tions to  the  form, — 

No  X  is  (2i  or  (2^  or  ...  or  Q^, 
then  combining  as  in  section  351,  and  finally  obverting  the 
result.     It  will  depend  on  the  form  of  the  original  proposi- 
tions whether  this  variation  is  desirable'. 

If  information  is  desired  with  regard  to  several  terms,  it 
may  be  found  convenient  to  bring  all  the  propositions  to  the 

form, — 

Everything  is  P^or  P^  ...  or  /*„; 

and  to  combine  them  at  once,  getting  in  a  single  proposition 
a  summation  of  all  the  information  given  by  the  separate 
propositions  taken  together.  From  this  we  may  immediately 
obtain  all  that  is  known  concerning  X  by  leaving  out  every 
alternative  that  contains  x,  all  that  is  known  concerning  Y 
by  leaving  out  every  alternative  that  contains  j,  and  so  on. 
The  following  may  be  taken  as  a  simple  example  of  the 
method.     It  is  adapted  from  Boole  {Laws  of  Thought,  p. 

118). 

''Given  ist,  that  wherever  the  properties  A  and  B  are 
combined,  either  the  property  C,  or  the  property  £>,  is 
present  also;  but  they  are  not  jointly  present:  2nd,  that 
wherever  the  properties  B  and  C  are  combined,  the  pro- 
perties A  and  P>  are  either  both  present  with  them,  or  both 

1  The  second  method  bears  a  somewhat  close  resemblance  to  Jevons's 
Indirect  Method ;  though  it  is  not  quite  the  same.  The  fust  method 
however  is  quite  distinct  from  Jevons's  method. 


344  COMPLEX  INFERENCES.  [part  iv. 

absent.    Shew  that  where  A  is  present,  either  ^  or  C  is 
absent." 

The  premisses  may  be  written, — 

AB  is  Cd  or  cD-,  (i) 

BC  is  AD  or  ad,  (2) 

Then,  we  may  immediately  obtain,— 

from  (i),  ^  is  <^  or  Cd  or  cD', 
and  from  (2),  ^  is  ^  or  ^  or  D'; 
therefore  (by  combining  these),  ^  is  ^  or  cD-, 

therefore,  ^  is  ^  or  r; 
which  is  the  desired  result. 

This  is  a  simple  example;  but  many  more  complicated 
ones  will  be  found  in  the  following  pages. 

The  method  here  described  will  I  think  in  nearly  every 
case  be  found  less  laborious  than  that  employed  by  Jevons^ 
—namely,  the  writing  down  all  the  possible  ^/r/W  alterna- 
tives so  far  as  the  terms  involved  are  concerned,  and  then 
striking  out  those  that  are  inconsistent  with  the  premisses. 
Also,  while  It  neither  requires    that  propositions  shall  be 
reduced  to  the  form  of  equations,  nor  involves  the  use  of 
mathematical  symbols  or  diagrams,  I  have  not  in  practice 
found  it  less  effective  than  the  methods  of  Bode  and  Venn^ 
^     I  shall  further  endeavour  to  shew  in  subsequent  sec- 
tions, how  special  results  may  frequently  be  obtained  in  a  still 
simpler  way  by  the  aid  of  various  formal  processes.    In  some 
of  the  examples  that  follow  both  the  general  method  and 
special  methods  are  employed. 

'  An  intermediate  step  might  be  introduced  here,  namely,  ABC  is  D 
»  Pure  Logic,  pp.  44,  45  ;  PHnciphs  of  Science,  chapter  vi 
At  the  same  time  of  course  these  methods  have  a  peculiar  interest 
and  significance  of  their  own. 


CHAP.  VII.]  COMPLEX  INFERENCES. 


345 


While  the  special  methods  are  as  a  rule  to  be  preferred 
when  they  have  been  discovered,  it  generally  takes  some 
time  and  ingenuity  to  discover  them.  On  the  other  hand, 
the  general  method  above  described  may  be  always  imme- 
diately applied  without  any  preliminary  study  of  the  case. 
Also,  while  special  methods  are  useful  to  establish  given 
results,  we  can  ordinarily  be  satisfied  that  we  have  a  com- 
plete solution  with  regard  to  any  term  only  when  we  have 
employed  the  general  method. 


CHAPTER  VIII. 

PROBLEMS    INVOLVING   THREE   TERMS. 

356.  Given  that  everything  is  either  Q  or  /v,  and 
that  all  R  is  Q,  unless  it  is  not  P,  prove  that  all  P 
is  Q. 

The  premisses  may  be  written, — 

r  is  e,  (i) 
PR  is  Q.  (2) 

By(i\PrkO. 
by  (2),    PRisQ; 

but  P  is  /V  or  TV?; 

therefore,   Pis  Q. 

357.  Where  A  is  present,  ^  and  C  are  cither  both 
present  at  once  or  absent  at  once;  and  where  dT  is 
present,  A  is  present.  Describe  the  class  not-B  under 
these  conditions.  [Jcvons,  Studies,  p.  204.] 

The  premisses  are, — 

A  is  BC  ox  be,  (i) 
C  is  A,  (2) 

By  (i),    Ab  is  <r, 
by  (2),  rt'/;  is  r, 
therefore,   /^  is  r. 

The  solution  is,  therefore,  "Where  B  is  absent,  Calso 
will  be  absent" 


CHAP,  viii.]         COMPLEX   INFERENCES. 


347 


358.  Given  (i)  P  is  QR,  (2)  /  is  qr\  shew  that 
(3)  Q  is  RP,  (4)  R  is  PQ. 

359.  Given  (i)  R  is  P  or  pq,  (2)  ^  is  7?  or  Pr, 
(3)  qR  is  P;  shew  that/  is  (2^. 

360.  Whenever  X  is  present,  F  and  Z  are  both 
present;  and  whenever  X  is  absent,  Fand  Z  are  both 
absent.  What  can  thence  be  inferred  with  regard  to 
the  relation  between  F  and  Z  t 

361.  (i)  Wherever  there  is  smoke  there  is  also 
fire  or  light ; 

(2)  Wlicrever  there  is  light  and  smoke  there  is 
also  fire : 

(3)  There  is  no  fire  without  either  smoke  or  light. 

Given  the  truth  of  the  above  propositions,  what  is 
all  that  you  can  infer  with  regard  to  (i)  circumstances 
where  there  is  smoke ;  (ii)  circumstances  where  there 
is  not  smoke ;  (iii)  circumstances  where  there  is  not 
light }  [W.] 

Let  A  =  circumstances  where  there  is  smoke, 
B  =  circumstances  where  there  is  light, 
C=  circumstances  where  there  is  fire. 

The  premisses  are, — 

^  is  ^  or  C,  (i) 

AB  is  C,  (2) 

C  is  ^  or  B.  (3) 

(i)  and  (2)  yield  A  is  C. 

By  (3)  ab  is  c;  therefore,  ^  is  ^  or  r. 

By  (i)  and  (3),  ^  is  d^  or  C,  and  also  A  or  c,  therefore, 
b  is  AC  or  ac. 


348  COMPLEX  INFERENCES.  [part  iv. 

We  have  then, — 

(i)  Where  there  is  smoke,  there  is  fire; 

(ii)  Where  there  is  not  smoke,  there  is  either  light  or 
there  is  no  fire; 

(iii)  Where  there  is  no  light,  there  is   either  both  fire 
and  smoke  or  neither  fire  nor  smoke. 

362.  Shew  the  equivalence  between  the  two  sets 
of  propositions, — • 

(i)  A  is  BC, 
Bis  AC, 
C  is  AB. 

(ii)  A  is  BCy 
a  is  be, 

B  is  AC,  C  is  AB,  give  by  contraposition  a  is  k'. 

a  is  be  gives  by  contraposition  B  is  A,  Cis  A;  and  since 
A  is  BC,  we  have  B  is  AC,  C  is  AB. 

363.  Shew  the  equivalence  between  the  following 
sets  of  propositions: — 

(i)     b  is  aC, 
c  is  aB ; 

(2)  A  is  BC, 
b  is  aC\ 

(3)  Ax^BC, 
c  is  aB, 

364.  Shew  the  equivalence  between  the  following 
sets  of  propositions  : — 

(i)  a  is  BC, 
b  is  AC, 
C  is  Ab  or  aB ; 


CHAP.  VIII.]        COMPLEX   INFERENCES. 


(2) 


349 


a  is  BC, 
B  is  Ac  or  aC, 
c  is  AB\ 
(3)     A  is  Be  or  bC, 
bis  AC, 
c  is  AB, 

365.  A  is  Be  or  3C,  ^  is  ^  C,  ^  is  ^^.  Shew  that 
all  the  information  given  by  the  combination  of  these 
propositions  is  also  given  by  the  propositions, — A  is  b 
or  e,  b  is  A,  cis  AB  or  ab. 

366.  Apply  the  method  of  solution  described  in 
section  355  to  ordinary  syllogisms  in  Barbara,  Cesare, 
Camenes. 

Barbara  has  for  its  premisses, — 

(i)  MisF, 
(2)  S  is  M. 
By  (i),  S  is  ///  or  F\ 
by  (2),  *S'  is  Afj 
therefore,  S  is  3fF; 
therefore,  S  is  F. 
Ccsare,  (i)  No /'is  J/, 

(2)  S  is  M. 
By  (i),  S  is  ;//  or/; 
by  (2),  S  is  M', 
therefore,  6*  is  Mp\ 
therefore,  iS  is  not  F. 
Camenes,  (i)  F  is  Af, 

(2)  NoAfisS. 
By  (i),  Sis  Movjf; 
by  (2),  S  is  ;//; 
therefore,  S  is  mp ; 
therefore,  No  6*  is  F. 


350 


COMrLEX   INFERENCES. 


[part  IV. 


367.  Assign  propositions  concerning  the  terms 
gem,  rich,  rare^  whose  aggregate  force  shall  be  such 
that  no  further  assertion  can  be  made  about  the  same 
terms  without  contradicting  the  propositions  assigned. 

Jevons  regards  any  two  or  more  propositions  as  incon- 
sistent when  they  involve  the  total  disappearance  of  any 
term,  positive  or  negative,  {Studies  in  Deductive  Logic,  p. 
i8i);  and  in  dealing  with  such  a  question  as  the  above,  it 
is  perhaps  a  convenient  criterion  to  take.  It  is  equivalent 
in  this  instance  to  the  assumption  that  there  exist  gems 
and  not  gems,  rich  things  and  not  rich  things,  rare  things 
and  not  rare  things.  It  is  an  assumption,  however,  and  as 
such  should  always  be  explicitly  stated,  when  made.  Com- 
pare Part  II.  Chapter  viii.  The  reason  why  it  is  convenient 
to  make  it  here  is  that,  (as  shewn  in  section  io6),  on  the 
supposition  that  All  S  is  P  does  not  itself  imply  the  exist- 
ence of  S,  All  Si^  P  and  No  S  is  P  are  either  inconsistent, 
or  between  them  deny  the  existence  of  S.  We  must 
therefore  exclude  the  latter  possibility,  if  we  wish  to  be 
able  to  say  definitely  that  two  such  propositions  are  in- 
consistent. 

The  given  question  is  now  solved  by  the  t/iree  proposi- 
tions : — 

(i)     All  gems  are  rich  and  rare ; 

(2)  All  rich  things  are  rare  gems ; 

(3)  All  rare  things  are  rich  gems. 

We  can  make  no  further  assertion  regarding  these  terms 
or  their  negatives  which  are  not  either  implied  by  the  above, 
or  else  inconsistent  with  them  j  since  the  only  classes  which 
they  allow  to  remain  are  rich  rare  gems  and  not  rich  not 
rare  not  gems. 


CHAP.  VIII.]        COMPLEX   INFERENCES.  351 

The  tic'o  propositions, — 

(i)     All  gems  are  rich  and  rare  ; 
(ii)     All  things  not  gems  are  neither  rich  nor  rare; 
also  afford  a  solution. 

The  equivalence  between  (i),  (2),  (3),  and  (i),  (ii),  has 
been  already  shewn  in  section  362. 

The  student  should  now  find  other  sets  of  propositions, 
not  equivalent  to  the  above,  which  also  afford  a  solution  of 
the  given  problem. 


\ 


CHAPTER  IX. 

PROBLEMS   INVOLVING    FOUR  TERMS. 

368.  Suppose  that  an  analysis  of  the  properties 
of  a  particular  class  of  substances  has  led  to  the 
following  general  conclusions,  viz.: 

1st,  That  wherever  the  properties  A  and  B  are 
combined,  either  the  property  C,  or  the  property  D,  is 
present  also  ;  but  they  are  not  jointly  present. 

2nd,  That  wherever  the  properties  B  and  C  are 
combined,  the  properties  A  and  D  are  either  both 
present  with  them,  or  both  absent. 

3rd,  That  wherever  the  properties  A  and  B  are 
both  absent,  the  properties  C  and  D  are  both  absent 
also ;  and  vice  vcrsd,  where  the  properties  C  and  D 
are  both  absent,  A  and  B  are  both  absent  also. 

Let  it  then  be  required  from  the  above  to  deter- 
mine what  may  be  concluded  in  any  particular  in- 
stance from  the  presence  of  the  property  A  with 
respect  to  the  presence  or  absence  of  the  properties 
B  and  C,  paying  no  regard  to  the  property  D, 

[Boole,  Laws  of  Thought,  pp.  118 — 120  ;  compare 
also  Venn,  Symbolic  Logic^  pp.  276 — 2^]%^ 


CHAP.  IX.]  COMPLEX   INFERExNCES.  353 

One  solution  has  already  been  given  in  section  355. 
We  might  also  proceed  as  follows.    The  premisses  are  : 


AB  is  Cd  or  cD, 
Bel's.  AD  ox  ad, 

ab  is  cd, 
cd  is  ab. 

By  (i).  No  AB  is  CD, 
therefore,  No  A  is  BCD. 

By(ii),  No^CTis^^, 
therefore,  No  A  is  BCd. 

Combining  (i)  and  (2),  we  have, — 

No  A  is  BC, 


(i) 

(ii) 
(iii) 

(iv) 
(i) 


/.  c,  All  ^  is  ^  or  c. 


This  solves  the  problem  as  set. 

Boole  also  shews  that  All  bC  is  A.  This  is  a  contra- 
positive  of  (iii).  We  have  not  required  to  make  use  of  (iv) 
at  all. 

369.  Given  the  same  premisses  as  in  the  preceding 
section,  prove  that : — 

(i)  Wherever  the  property  C  is  found,  either 
the  property  A  or  the  property  B  will  be  found  with 
it,  but  not  both  of  them  together. 

(2)  If  the  property  B  is  absent,  either  A  and  C 
will  be  jointly  present,  or  C  will  be  absent. 

(3)  If  A   and   C  are  jointly  present,  B  will  be 

[Boole,  Laws  of  Thought,  p.  129.] 

By  (i),^^Cis^, 

by(ii),  ^^CisZ>; 

i.e.,  there  is  no  such  thing  as  ABC, 

i.  e.,  Cis  ^  or  b. 

K.  L.  23 


absent. 
First, 


354 


COMPLEX  INFERENCES.  [part  iv. 


Also,  by  contraposition  of  (iii),  C  is  ^  or  ^j 

therefore,  C  is  Ab  or  aB,  (i) 

Seco7idI}\  By  (iii),  h'vs^  A  ox  c, 

therefore,  b  \s  AC  or  c.  (2) 

Thirdly,  We  have  shewn  that  it  follows  from  (i)  and  (ii) 
that  there  is  no  such  thing  as  ABC, 

therefore,  AChb.  (3) 

370.  It  is  known  of  certain  things  that  (i)  where 
the  quality  A  is,  B  is  not ;  (2)  where  B  is,  and  only 
where  B  is,  C  and  D  are.  Derive  from  these  con- 
ditions a  description  of  the  class  of  things  in  which  A 
is  not  present,  but  C  is.     [Jevons,  Studies,  p.  200.] 

The  premisses  are, — 

(i)  A  is  b\ 

(2)  Bis  CD) 

(3)  CD  is  B. 

(i)  affords  no  information  with  regard  to  aC.  Cf  sec- 
tion 353. 

But  by  (2),  ^Cis^orZ>; 

and  by  (3),  aC  '\s  B  ox  d) 

therefore,  aC  is  BD  or  bd. 

371.  Taking  the  same  premisses  as  in  the  pre- 
vious section,  draw  descriptions  of  the  classes  Ac,  ab, 
and  cD.  [Jevons,  Studies,  p.  244.] 

We  have  (i)     A  is  b; 

(2)  Bis  CD', 

(3)  CD  is  B. 

By  (i),  ^^is  ^; 
by  (2),  Ac  is  b; 
(3)  affords  no  information  with  regard  to  Ac, 


CHAP.  IX.]  COMPLEX   INFERENCES. 


355 


^y  (3)>  ^^  is  c  or  d. 

By  (i),  cD  is  a  ox  b; 
by  (2),  cD  is  b  ; 
therefore,  cD  is  b. 

We  can  obtain  no  farther   information  with  regard  to 
ab  and  cD, 

The  desired  results,  therefore,  are, — 

Ac  is  b ; 
ab  is  c  or  d', 
cD  is  b. 

372.  There  is  a  certain  class  of  things  from  which 
A  picks  out  the  '  X  that  is  Z,  and  the  Y  that  is 
not  Z;  and  B  picks  out  from  the  remainder  '  the  Z 
which  is  Y  and  the  X  that  is  not  K'  It  is  then 
found  that  nothing  is  left  but  the  class  '  Z  which  is 
not  X!  The  whole  of  this  class  is  however  left. 
What  can  be  determined  about  the  class  originally  } 

[Venn,  Symbolic  Logic,  pp.  26^,  8.] 

The  chief  difficulty  in  this  problem  consists  in  the  accu- 
rate statement  of  the  premisses.  Call  the  original  class  W. 
We  then  have, — 

Wis  XZ  or  Yz  or  YZ  or  Xy  or  xZ,  (i) 

^Zis  W.  (2) 

No  xZ  is  WXZ  or  IVYz  or   IVYZ  or    WXy, 
i.e.,  (leaving  out  such  part  of  this  statement  as  is  merely 
identical), 

No^Zis  WYZ,  (3) 

We  may  now  proceed  as  follows  : — 
%(3),  No  WisxYZ',  (4) 

By  (i),  ^oWisxyz.  (5) 

23—2 


356 


COMPLEX   INFERENCES. 


[part  IV. 


Combining  this  with  (4),  we  find  that  the  class  did  not 
originally  contain  any  not- A"  that  was  either  both  V  and  Z 
or  neither  V  nor  Z. 

(2)  affords  no  information  regarding  the  class  JV,  except 
that  everything  that  is  Z  but  not  X  is  contained  within  it. 
The  student  may  however  notice  that  from  this  proposition 
in  conjunction  with  (3),  it  may  be  deduced  that  all  VZ 
isX 

373.  At  a  certain  town  where  an  examination  is 
held,  it  IS  known  that, 

(i)  Every  candidate  is  either  a  junior  who  does 
not  take  Latin,  or  a  Senior  who  takes  Composition. 

(2)  Every  junior  candidate  takes  either  Latin  or 
Composition. 

(3)  All  candidates  who  take  Composition,  also 
take  Latin,  and  arc  juniors. 

Shew  that  if  this  be  so  there  can  be  no  candidates 
there.  [Venn,  Symbolic  Logic,  pp.  270,  i.] 

Let  X  -  candidate, 

A  =  junior,  so  that  a  =  senior, 
J^  =  taking  Latin, 
C  =  taking  Composition. 

We  then  have, — 

XisAb  or  aC;  (i) 

X^is^orC;  (2) 

XC  is  AB.  (3) 

(2)  and  (3)  give  XA  is  B ; 
therefore,  No  AT  is  Ad  ; 
also  by  (3),  No  X  h  aC, 

It  therefore  follows  from  (i)  that  there  can  be  no  such 
thing  as  X. 


CHAP.  IX.]  COMPLEX   INFERENCES.  357 

374.  Given  (i);i' IS  j^^;  (2)ZW  isy;  (j)  y  is  ZW; 
(4)  ^  is  jy ;  (5)  XIV  is  VZ ;  shew  that  (i)  Nothing  is 
W,  (ii)  Everything  is  XVZ. 

[Venn,  Symbolic  Logic,  pp.  271,  2.] 

By  (5),  XJFis  YZ', 
but  by  (2),  No  ^is  KZ; 
therefore.  Nothing  is  XW, 

By  (i),  X  JV is yz ; 
but  by  (3),  Nothing  is  yz ; 
therefore.  Nothing  is  x  W, 

But  WisXlVorxW; 
therefore,  Nothing  is  W.  (i) 

By  {i),x\syz] 
but  by  (3),  Nothing  is^^; 
therefore.  Nothing  is  x. 

By  (3),^  is  IV, 

and  by  (4),  z  is  W; 
but  by  (i).  Nothing  is  IV; 
therefore,  Nothing  is  j^  or  j?  ; 
therefore.  Nothing  h  x,  y  ox  z ; 
i.e.,  Everything  is  XVZ.  (ii) 

375.  If  thriftlessness  and  poverty  are  inseparable, 
and  virtue  and  misery  are  incompatible,  and  if  thrift 
be  a  virtue,  can  any  relation  be  proved  to  exist  be- 
tween misery  and  poverty  }  If  moreover  all  thriftless 
people  are  either  virtuous  or  not  miserable,  what 
follows }  [v.] 

Let     A  =  thriftless, 
B  =  poor, 
C=  virtuous, 
L>  =  miserable. 


358 


COMPLEX   INFERENCES. 


[part  IV. 


Then  the  premisses  as  first  given  may  be  written, — 

A  is  B\  (i) 

B\sA\  (2) 

No   C'lsD,  (3) 

a  is  C.  (4) 

Can  we  now  find  any  relation  between  B  and  D  ? 

We  may  proceed  by  finding  all  that  we  can  assert  con- 
cerning B  and  D  respectively. 

By  (2),  Bis  A; 
by  (3),  B  is  c  or  d* ; 
therefore,  B  is  Ac  or  Ad; 
therefore.  If  B  is  D,  it  is  Ac ; 

i.e.,  If  poverty  is  accompanied  by  misery,  it  is  also  accom- 
panied by  thriftlessness,  and  it  is  not  accompanied  by 
virtue. 

Again,  by  (i),  B>  is  a  or  B ; 

by  (2),  Z>  is  A  or  d'j 
therefore,  Z)  is  AB  or  ab ; 

by  (3),  D  isc, 
therefore,  D  is  A  Be  or  ahc ; 
by  (4),  D  is  A  or  Cj 
therefore,  D  is  ABc\ 
i.e.,  Misery  is  always  accompanied  by  poverty ;  or,  misery 
is  never  found  unaccompanied  by  poverty. 

This  result  might  also  be  obtained  by  two  ordinary 
syllogisms  in  Barbara,  as  follows ; 

^  If  we  adopted  the  equational  rendering  of  propositions,  which 
however  I  have  intentionally  avoided,  "^  is  />'"  and  '^ B  \%  A  "  would 
of  course  be  summed  up  in  "^  =-5."  In  cases  of  this  kind,  the  equa- 
tional rendering  is  at  its  best. 

2  (i)  gives  no  information  regarding  B\  and,  so  far  as  B  is  con- 
cerned, (4)  merely  repeats  part  of  the  information  given  by  (2). 


CHAP.  IX.]  COMPLEX   INFERENCES.  359 

By  (3),  D  IS  c\ 

by  (4),  c  is  A\ 
therefore,  Z>  is  ^ ; 

by  (i),  A  is  B; 
therefore,  Z>  is  B, 

If  to  the  given  premisses  we  now  add  (5)  ^  is  C  or  d, 
we  find  that  Z>  is  both  A  and  a,  a  result  which  must  be 
interpreted  as  affirming  the  non-existence  of  Z> ;  i.e.,  There 
is  no  such  thing  as  misery. 

It  will  be  a  more  complete  answer  to  the  latter  part  of 
the  question  to  note  the  full  result  of  combining  all  the  given 
premisses. 

By  (i),  Everything  is  ^  or  ^ ; 
by  {2),  Everything  is  ^  or  ^ ; 
therefore,  Everything  is  ^^  or  ab; 
by  (3),  Everything  is  c  or  d; 
therefore.  Everything  is  ABc  or  ABd  or  abc  or  abd ; 
by  (4),  Everything  is  A  or  C; 
therefore,  Everything  is  ABc  or  ABd  or  abCd; 
by  (5),  Everything  is  ^  or  C  or  ^; 
therefore,  Everything  is  ABd  or  abCd. 

This  gives  us  : — 

A  is  Bd; 

a  is  bCd; 

B  is  Ad; 

b  is  aCd; 

C  is  ABd  or  abd; 

c  is  ABd; 

There  is  no  such  thing  as  D ; 

d  is  AB  or  abC. 


36o 


COMPLEX  INFERENCES.  [part  iv. 


376.  A  given  class  is  made  up  of  those  who  are 
not  male  guardians,  nor  female  ratepayers,  nor  lodgers 
who  are  neither  guardians  nor  ratepayers.  How  can 
we  simplify  the  description  of  this  class  if  we  know 
that  all  guardians  are  ratepayers,  that  every  person 
who  is  not  a  lodger  is  either  a  guardian  or  a  rate- 
payer, and  that  all  male  ratepayers  are  guardians  ? 

[v.] 

Let  -X'=the  given  class, 

A  -  male, 
B  -  guardian, 
C=  ratepayer, 
D  =  lodger. 

Then  X  is  made  of  those  who  are 

not  AB  nor  aC  nor  bcD ; 
that  is,  X  is  made  up  of  those  who  are 

aBc  or  AdC  or  aai  or  Al^d  or  da/. 

But  we  are  told  that, — 

(i)    B  is  C; 

(2)  d  is  B  ox  C; 

(3)  ACisB, 

From  (i),  it  follows  that  there  is  no  aB^;  from  (2),  that 
there  is  no  hd;  from  (3),  that  there  is  no  Al?C;  from  (2) 
and  (3)  taken  together  that  there  is  no  A/?d;  from  (i)  and 
(2)  taken  together  that  there  is  no  acd. 

It  therefore  follows  that  the  given  class  is  itself  non- 
existent. 

We  might  arrive  at  the  same  result  as  follows  : — 
By  (i),  Everything  is  /^  or  C; 
by  (2),  Everything  is  i?  or  C  or  Z); 
therefore,  Everything  is  bD  ox  C) 


CHAP.  IX.]  COMPLEX   INFERENCES. 


361 


by  (3),  Everything  is  ^  or  ^  oxc\ 
therefore,  Everything  is  abD  or  bcD  ox  aC  or  BC\ 
but  abD  is  ^C  or  bcD  ^, 
and  BC  is  AB  or  aC^-, 
therefore,  Everything  is  ^C  or  bcD  or  AB\ 

which  again  shews  that  the  given  class  is  non-existent. 

377.  Given  that  everything  that  is  Q  but  not  5  is 
either  both  P  and  R  or  neither  P  nor  R  and  that 
neither  R  nor  ^  is  both  P  and  Q,  shew  that  no  P 

378.  Where  C  is  present,  A^  B  and  D  are  all 
present ;  where  D  is  present,  A^  B  and  C  are  either 
all  three  present  or  all  three  absent.  Shew  that 
when  either  A  or  ^  is  present,  C  and  D  are  either 
both  present  or  both  absent.  How  much  of  the 
given  information  is  superfluous  so  far  as  the  desired 
conclusion  is  concerned  ? 

379.  Every  voter  is  both  a  ratepayer  and  oc- 
cupier, or.  not  a  ratepayer  at  all. 

If  any  voter  who  pays  rates  is  an  occupier,  then 
he  is  on  the  list. 

No  voter  on  the  list  is  both  a  ratepayer  and  an 
occupier. 

Examine  the  results  of  combining  these  three 
statements.  [v.] 

380.  At  a  certain  examination,  all  the  candidates 
who  were  entered  for  Latin  were  also  entered  for  either 


^  Since,  by  the  law  of  Excluded  Middle,  abD  is  abCD  or  ahcD. 
2  Since,  by  the  law  of  Excluded  Middle,  BC  is  ABC  or  aBC. 


362 


COMPLEX   INFERENCES. 


[part  IV. 


Greek,  French,  or  German,  but  not  for  more  than  one 
of  these  languages  ;  all  the  candidates  who  were  not 
entered  for  German  were  entered  for  two  at  least  of 
the  other  languages ;  no  candidate  who  was  entered 
for  both  Greek  and  French  was  entered  for  German, 
but  all  candidates  who  were  entered  for  neither 
Greek  nor  French  were  entered  for  Latin.  Shew 
that  all  the  candidates  were  entered  for  two  of  the 
four  languages  but  none  for  more  than  two. 

381.  AB  is  D,  ab  is  cd,  c  is  ABD  or  ahd,  D  is 
AB,  All  the  information  given  by  these  propositions 
is  also  given  by  the  propositions, — ABC  is  D^  abd  is 
Cy  c  is  AD  or  abdy  D  is  A B  or  ac  or  Bc)  and  vice 
versa, 

382.  Shew  that  the  following  sets  of  propositions 
are  equivalent  : — 

(i)  aisborc'yb  is  aCd\  c  is  aB  \  D  is  c, 

(2)  A  is  BC\  b  is  aC\  C\s  ABd  ox  abd. 

(3)  A  \s  B  \  B  \s  A  ox  c\  CIS  aB  ;  D  \s  c, 

(4)  b  is  aC\  A  is  C;  C'ls  d;  aC  is  b. 

(5)  c  is  aB  ;  D  is  aB  \  A  \s  B  \  aB  is  c. 

(6)  A  is  BC\  BC  is  A;  D  is  Be  ;  bis  C. 


CHAPTER  X. 


PROBLEMS   INVOLVING   FIVE  TERMS. 


383.  Let  the  observation  of  a  class  of  natural 
productions  be  supposed  to  have  led  to  the  follov/ing 
general  results. 

1st.  That  in  whichsoever  of  these  productions  the 
properties  A  and  C  are  missing,  the  property  E  is 
found,  together  with  one  of  the  properties  B  and  Z>, 
but  not  with  both. 

2nd.  That  wherever  the  properties  A  and  D  are 
found  while  E  is  missing,  the  properties  B  and  C 
will  either  both  be  found,  or  both  be  missing. 

3rd.  That  wherever  the  property  A  is  found  in 
conjunction  with  either  B  or  E,  or  both  of  them,  there 
either  the  property  C  or  the  property  D  will  be  found, 
but  not  both  of  them.  And  conversely,  wherever  the 
property  C  ox  D  is  found  singly,  there  the  property  A 
will  be  found  in  conjunction  with  either  B  or  E,  or 
both  of  them. 

Shew  that  it  follows  that  In  whatever  substances 
the  property  A  is  fonnd,  there  zvill  also  be  fonnd  cither 
the  property  C  or  the  property  D,  but  not  both,  or  else  the 


364  COMPLEX  INFERENCES.  [part  iv. 

properties  B,  C,  and  Z>,  will  all  be  ivanting.  And  con- 
versely, where  either  the  property  C  or  the  property  D  is 
found  singly,  or  the  properties  B,  C,  and  D,  are  together 
missing,  there  the  property  A  zuill  be  found. 

[Boole,  Laws  of  Thought,  pp.  146—148.  Cp.  also 
Venn,  Symbolic  Logic,  pp.  280,  281.] 

The  premisses  are  as  follows : — 

ist,  All  ac  is  BdE  or  IDE-,  (i) 

2nd,  All  ADe  is  BC  or  he,  (ii) 

3rd,  All  AB  is  Cd  or  cD;  (iii) 

All  AE  is  Cd  or  cD)  (iv) 

All  Cd'isAB  or  AE;  (v) 

All  cD  is  AB  or  AE.  (vi) 
We  are  required  to  prove: — 

All  A  is  Cd  or  cD  or  bed;  (a) 

All  CVis  A;  {(B) 

AW  cD  IS  A;  (y) 

All  bed  is  A,  (8) 

Eirst,  By  (iii)  and  (iv),  If  A  Is  B  or  E  it  is  G/  or  eD; 

therefore,  A  is  Cd  or  rZ)  or  ^t%  (i) 

By  (ii),  Ae  is  either  BC  or  /^r  or  d; 

therefore,  Abe  is  /i^  or  d; 
therefore,  Abe  is  bee  or  /;^/^.  (2) 

By  (v),  Cd  is  B  or  E; 
therefore,  C  is  B  or  L>  or  E; 
therefore  (by  contraposition),  bde  is  c; 

therefore,  bde  is  bed; 
therefore.  If  Abe  is  bde  it  is  bed.  (3) 

Again  by  (vi),  eL>  is  i>  or  E; 
therefore  (as  above),  bee  is  d; 
therefore,  If  Abe  is  bee  it  is  ^^^/.  (4) 


CHAP.  X.]  COMPLEX   INFERENCES.  365 

Therefore,  by  (2),  (3),  and  (4),  Abe  is  bed; 
therefore  from  (i),  ^  is  either  Cd  or  eD  or  bed.        (a) 

Secondly,  (fi)  and  ^7)  follow  immediately  from  (v)  and  (vi). 

Thirdly,  from  (i),  we  have  directly,  No  ac  is  bd; 

therefore  (by  conversion).  No  bed  is  a; 


therefore,  All  bed  is  A. 


(8) 


The  first  of  the  desired  results  might  also  be  obtained  as 
follows : — 

As  before  we  may  shew  that 

A  is  Cd  or  cD  or  be; 
and  we  therefore  have  what  is  required  if  w^e  can  shew  that 

Abe  is  cd. 
By  (ii),  Abe  is  c  or  d; 
by  (v),  Abe  is  c  or  D; 
therefore,  Abe  is  c; 
by  (vi),  Abe  is  C  or  d; 
therefore,  Abe  is  r^. 

We  have  here  employed  a  modification  of  the  general 
method  described  in  section  355. 

We  might  also  by  this  method  obtain  a  complete  solution 
of  the  problem  so  far  as  A  is  concerned. 

(i)  gives  no  information  whatever  with  regard  to  A  \ 
But  by  (ii),  A  is  BC  or  be  or  d  or  E; 
by  (iii),  ^  is  ^  or  Cd  or  cD; 
therefore,  A  is  Cd  or  be  or  bd  or  bE  or  cDE; 

by  (iv),  A  is  Of  or  ^/?  or  e; 
therefore,  A  is  Cd  or  cZ>^  or  /'(t/?  or  bee  or  /^^<?; 
by  (v),  ^  is  ^  or  ^  or  ^  or  D; 
therefore,  A  is  cDE  or  ^<rZ>  or  bee  or  ^G/  or  C^^"; 

1  Since  a  appears  in  the  subject.     Cf.  section  353. 


366 


COMPLEX   INFERENCES. 


[part  IV. 


by  (vi),  ^  is  ^  or  i?  or  (7  or  d) 
therefore,  A  is  cDE  or  bcde  or  BCd  or  CdE\ 

This  includes  the  partial  solution  with  regard  to  A, — 

A  is  Cd  or  cD  or  bed. 
Boole  contents  himself  with  this  because  he  has  started 
with  the  intention  of  eliminating  E  from  his  conclusion. 

384.  Given  the  same  premisses  as  in  the  preceding 
section,  prove  that  If  the  property  A  is  absent  and  C 
present^  D  is  pj'esent. 

[Boole,  Lazvs  of  Thought,  p.  148.] 

By  (v),  Cd  is  A; 
therefore  (by  contraposition),  aCis  E>. 

385.  Given  the  same  premisses  as  in  section  383, 
shew  that, — 

1st.  If  the  property  B  he  present  in  one  of  the  pro- 
ductions, cither  the  properties  A,  Q  and  D,  are  all 
absent,  or  some  one  alone  of  them  is  absent.  And  con- 
versely, if  they  are  all  absent  it  may  be  concluded  that 
the  property  B  is  present, 

2nd.  If  A  and  C  are  both  present  or  both  absent, 
D  will  be  absent,  quite  independently  of  the  presence  or 
absence  of  B,  [Boole,  Lazus  of  Thought,  p.  149.] 

^  To    shew    that    this    method    is    not    very    laborious,    it    may 
perhaps  be  worth  mentioning  that  no  step  in  my  original  working  is 
omitted  in  the  above.     In  the  first  instance,  without  any  knowledge  of 
the  solution  that  would  result,  I  obtained  it  by  aid  of  the  steps  here 
inserted  without  erasure  or  rough  working  of  any  kind.     I  am  doubtful 
whether  by  any  other  method  the  result  could  be  reached  more  expe- 
ditiously.    It  is  probable  that  at  first  the  student  may  require  to  insert 
some  other  steps  in  the  reasoning,  and  that  the  possible  simplifications 
may  not  immediately  occur  to  him.     But  this  will  be  remedied  by  a 
very  little  practice. 


CHAP.  X.]  COMPLEX  INFERENCES. 


Z^l 


We  have  to  shew, — 

(a)     B  is  acd  or  aCD  or  AcD  or  ACd; 

(/?)    acd  is  B; 
(y)     ^  C  is  ^; 
(8)     ac  is  d. 
First,  By  (iii),  B  is  Cd  or  cD  ox  a; 

therefore,  B  is  A  Cd  or  AcD  or  a.  (i) 

By  (i),  ac  is  BdoxbD; 
therefore,  No  ac  is  BD) 
therefore,  No  aB  is  cD,  (2) 

By  (v),  C^  is  ^ ; 

therefore  (by  contraposition),  ^  is  ^  or  Z>; 

therefore.  No  a  is  Cd; 

therefore,  No  aB  is  Cd.  (3) 

By  (2)  and  (3),  No  aB  is  cD  or  Cd\ 

therefore,  All  aB  is  cd  or  CD. 

Combining  this  result  with  (i),  we  have, — 

Bis  A  Cd  or  AcD  or  acd  or  a  CD.  (a) 

Sccoftdly,  From  (i)  we  have  directly,  acd  is  BdE ; 

therefore,  acd  is  B.  (P) 

Thirdly,  By  (ii).  No  ADe  is  bC -, 

therefore,  No  ^C  is  hDe.  (i) 

By  (iii).  No  AB  is  CD ; 
therefore.  No  ^  C  is  BD.  (2) 

By  (iv)  No  AE  is  CD ; 
therefore,  No  ^  C  is  DE.  (3) 

Therefore,  by  (i),  (2),  and  (3),  \i  AC  is  Di\.  is  neither 
be  nor  B  01  E)  but  (by  the  law  of  excluded  middle), 
All  AC  is  either  B  ox  E  oxbe\ 
therefore,  No  ^C  is  Z> ; 
therefore,  All  ^C  is  ^.  (7) 


368  COMPLEX  INFERENCES.  [part  iv. 

Lastly,  By  (vi),  €D\'s,  A  ; 

therefore  (by  contraposition),  ac  is  d,  (S) 

For  complete  solutions  with  regard  to  B,  acd,  A  C,  ac, 
see  the  following  section. 

386.  Givxn  the  same  premisses  as  In  section  383, 
obtain  complete  solutions  with  regard  to  B,  acd,  AC, 
ac. 

Complete  solutions  with  regard  to  B,  acd,  AC,  ac,  may 
be  obtained  by  the  general  method  described  in  section  355 
as  follows  : — 

jFirsf,  By  (i),  B  is  dE  ox  A  or  C; 

by  (ii),  ^  is  C  or  a  or  d  or  E  ] 

therefore,  ^  is  C  or  dE  or  Ad  or  AE ; 

by  (iii),  B  is  Cd  or  cD  or  a ; 

therefore,  B  is  Cd  or  a  C  or  adE  or  AcDE\ 

(iv)  gives  no  information  with  regard  to  B  that  is  not 
already  given  by  (iii)  ; 

by  (v),  B  is  A  or  c  or  Z>; 

therefore,  B  is  AcDE  or  ACd  or  aCD  or  acdE, 

(vi)  gives  no  further  information  with  regard  to  B. 

The  above  includes  the  special  solution  given  by 
Boole, — 

B  is  acd  or  aCD  or  AcD  or  ACd. 

Secondly,  By  (i)  acd  is  BE. 

None  of  the  other  propositions  give  any  information 
with  regard  to  acd\  This  then  is  the  complete  solution  so 
far  as  acd  is  concerned. 


1  c:; 


Since  the  subjects  of  all  these  propositions  contain  either  Ay  C, 
or  D,     Cf.  section  353. 


CHAP.  X.] 
Thirdly, 


COMPLEX   INFERENCES. 


369 


By  (ii),  AC  is  B  or  d  or  E\ 

by  (iii),  ACis  b  or  d; 
therefore,  ACis  d or  bE ; 

by  (iv),  ^  C  is  ^  or  ^ ; 
therefore,  AC  is  d; 
by  {v),AC  is  B  or  E  or  B; 
therefore,  ACis  Bd or  dE. 

Lastly,  By  (i),  ac  is  BdE  or  IDE ; 

by  (vi),  ac  is  d ; 
therefore,  ac  is  BdE, 

387.  Every  A  is  one  only  of  the  two  B  or  C,  D 
is  both  B  and  C  except  when  B  is  E  and  then  it  is 
neither ;  therefore  no  A  is  D.  [De  Morgan.] 

This  example,  originally  given  by  De  Morgan,  (using 
however  different  letters),  and  taken  by  Professor  Jevons  to 
illustrate  his  symbolical  method,  {Principles  of  Science,  Vol.  i, 
p.  117;  Studies  in  Deductive  Logic,  p.  203),  is  chosen  by 
Professor  Groom  Robertson  to  shew  that  "the  most  com- 
plex problems  can,  as  special  logical  questions,  be  more 
easily  and  shortly  dealt  with  upon  the  principles  and  with 
the  recognised  methods  of  the  traditional  logic "  than  by 
Jevons's  system. 

"  The  mention  oi  E  as  E  has  no  bearing  on  the  special 
conclusion  A  is  not  D  and  may  be  dropt,  while  the  impli- 
cation is  kept  in  view ;  otherwise,  for  simplification,  let  BC 
stand  for  *  both  B  and  CJ  and  be  for  *  neither^  nor  C 
The  premisses  then  are, — 

(i)     Z>  is  either  ^(7  or /^^, 
(2)     A  is  neither  BC  nor  be, 
which  is  a  well-recognised  form  of  Dilemma  with  the  con- 
clusion A  is  not  D,     Or,  by  expressing  (2)  as  A  is  not 
K.  L.  24 


370 


COxMPLEX  INFERENCES. 


[part  IV. 


either  BC  or  be,  the  condiision  may  be  got  in  Camestres. 
If  it  be  objected  that  we  have  here  by  the  traditional  pro- 
cesses got  only  a  special  conclusion,  it  is  a  sufficient  reply 
that  any  conclusion  by  itself  must  be  special.  What  other 
conclusion  from  these  premisses  is  the  common  logic  power- 
less to  obtain?"     {Muid,  1876,  p.  222.) 

The  solution  is  also  obtainable  as  follows, — 
By  the  first  premiss,  A  is  Be  or  bC,  and  by  the  second, 
A  is  BC  or  be  or  d ; 

therefore,  A  is  Bed  or  bCd, 
therefore,  A  is  d. 

Professor  Robertson's  solution  is  in  this  case  preferable. 
But  I  append  the  above  as  a  further  illustration  of  my  own 
method.  Compared  with  the  problem  of  Boole's  just  dis- 
cussed or  with  the  problems  that  follow,  this  of  De  Morgan's 
is  not  particularly  complex. 

388.     Suppose  it  known  that, — 

(i)  Where  B  is  present  and  C  absent,  either  D  is 
present  or  E  is  absent; 

(2)  Where  A  and  D  are  present  and  C  absent,  B 
is  present ; 

(3)  Where  B  is  absent  and  C  present,  A  is 
present ; 

(4)  Where  C  and  D  are  present,  A  is  absent  or 
B  is  present ; 

(5)  Where  E  is  present  and  D  absent,  A  and  C 
are  not  both  present  nor  are  B  and  C  both  absent ; 

(6)  Where  B  is  present  and  D  absent,  C  is 
absent ; 

(7)  Where  A  is  present  and  E  absent,  either  B 
or  D  is  present ; 


CHAP.  X.]  COMPLEX   INFERENCES.  371 

then  we  can  shew  that, — 

(i)      Where  A  is  present,  B  is  present ; 

(ii)     Where  B  is  absent,  C  is  absent ; 

(iii)    Where  C  is  present,  D  is  present ; 

(iv)   Where  D  is  absent,  E  is  absent. 

First,  By  (2),  AeD'x's.B) 

by  (4),  CD  is  a  or  B ; 

therefore,  ACB  is  B; 

therefore,  AB>  is  B, 

Again,  by  (5),  No  ^^  is  ^C; 

therefore,  AdE  is  e ; 

therefore,  AdE  is  edE. 

But  also  by  (5),  edE  is  B; 
therefore,  AdE  is  B ; 
by  (7),  Ade  is  B; 
therefore,  Ad  is  B. 

But  we  have  shewn  above  that  AE>  is  B ; 

therefore,  A  is  B,  (i) 

Secondly,  By  (3),  bC is  A; 

therefore,  ab  is  e. 

By  (i),  b  is  a; 

therefore,  ^  is  ^^ ; 

therefore,  b  is  c,  (ii) 

T/i/rdfy,  By{6),  Bd  is  e; 

therefore,  BCis  D, 

By  (ii),  CisB; 
therefore,  Cis  BC', 
therefore,  C  is  D,  (iii) 

Fourthly,  By  (\\  Beis  D  01  e'y 

therefore,  Bed  is  e, 

24 — 2 


372 


COMPLEX  INFERENCES. 


[part  IV. 


By  (5),  No  dE  is  bc\ 
therefore,  bed  is  ^  ; 
therefore,  cd  is  e. 

By  (ill),  d  is  cd-j 
therefore,  d  is  e.  (iv) 

389.     Given  that,— 

(i)     Where  A  and  C  are  absent,  D  and  E  are 
absent ; 

(2)  Where  B  and  Z>  are  present,  C  is  present  or 
else  A  is  present  and  E  absent ; 

(3)  Where  B  is  absent,  ^  and  C  arc  present  or  yl 
and  D  are  absent ; 

(4)  Where  C  and  ^  are  present,  Ay  B,  and  D  are 
absent ; 

(5)  Where  D  is  absent  and  E  is  present,  -^4  is 
absent ; 

(6)  Where  ^  and  E  are  absent,  either  A  or  D  is 
absent. 

Prove  that, — 

(i)     Where  A  is  present  and  B absent,  Cis  present; 
(ii)     Where  B  is  absent  and  D  present,  A  and  C 
are  present  and  E  is  absent ; 

(iii)     Where  C  is  absent,  D  and  E  are  absent ; 

(iv)     Where  E  is  present,  A,  B,  and  Z^  are  absent 
and  C  is  present. 

First,  By  (3),  ^  is  -<4C  or  tz//; 

therefore,  Ab  is  C  (i) 

Secondly y  By  (3),  ^  is  ^^  (7  or  a'^  ; 

therefore,  bD  is  ^  C 


CiiAP.x.]  COMPLEX  INFERENCES. 


373 


By  (4),  CE  is  abd ; 
therefore.  Everything  is  r  or  ^  or  abd ', 
therefore,  bD  \s  e  01  e] 
but  as  shewn  above,  bD  is  ^C; 
therefore,  bD  is  A  Ce. 


00 


Thirdly, 


By  (i),  fl'^  is  de\ 
therefore,  ^  is  y^  or  dc. 


By  (2).  ^Z^  is  Cor^^; 

therefore.  Everything  is  /^  or  ^  or  C  or  Ac ; 
therefore,  r  is  ^  or  ^  or  ^^ ; 
therefore,  e  is  Ae  or  Ab  or  ^^  or  ^<?. 

By  (3),  ^  is  ^Cor  ^^; 

therefore,  cis  B  ot  ad; 

therefore,  e  is  ^^^  or  ABd  or  ^^. 

By  (5),  dE  is  a; 

therefore,  ^  is  Z>  or  ^  or  ^; 

therefore,  e  is  ABe  or  ^^; 

by  (6),  eis  a  or  d  or  E ; 

therefore,  e  is  ^i?. 


(iii) 


Fourthly,  By  (i),  ^  is  ^  or  C; 

By  (2),  Z'is  /^  or  ^or  C) 
therefore,  E  is  ^^  or  Ad  or  C; 
by  (3),  E  is  B  or  A  Cor  ad; 
therefore,  Z:  is  ^C  or  ^^^  or  BC  or  ^ G/; 

by  (4),  E  is  fl'^^  or  e ; 
therefore,  E  is  A  Bed  or  a<^G/; 

by  (5),  Z"  is  flJ  or  Z>; 

therefore,  E  is  a^CV.  (iv) 


374  COMPLEX  INFERENCES.  [part  IV. 

390.  Given  that,— 

ad  Is  d  or  e; 

b  is  AcD  or  ad  or  cdc ; 

c'ls  A  bD  or  bde ; 

c  is  aBCd  ox  bcd\ 
shew  that,  A  is  ^C^  or  bcDE  or  <5^^^ ; 

a  is  BCDE  or  ^C</^  or  ^G/Zi  or  ^^^^ ; 

i5  is  ^  CE  or  ^^^^^^  or  CDE ; 

C  is  ^^jIS'  or  aBde  or  ^^^^^  or  BDE, 

391.  (i)     Where  /*  is  present  while  Q  and  R  are 
absent,  5  and  T  are  present. 

(ii)     2  and  R  are  always  present  or  absent  to- 
gether. 

(iii)     Where  <2,  R  and  5  are  all  present,  P  and  T 
are  either  both  present  or  both  absent. 

(iv)     Where  Q  and  R  are  present  while  5  is  ab- 
sent, P  is  present  and  T  is  absent 

(v)     Where  ^  is  present  while  /*,  Q  and  R  are  all 
absent,  T  is  present. 

Then, — 

(i)     Where  P  is  present  while  Q  is  absent,  5  and 
T  are  present  and  R  is  absent. 

(2)  Where  P  is  present  while  ^  is  absent,  Q  and 
/l  are  present  and  T  is  absent. 

(3)  Where  P  is  absent  while  Q  is  present,  R  and 
6*  are  present  and  T  is  absent. 

(4)  Where  P  is  absent  while  T  is  present,  0  and 
R  are  both  absent. 


CHAP.  X.] 


COMPLEX   INFERENCES. 


375 


(5)  Where  Q  is  present  while  vS"  is  absent,  P  and 
R  are  present  and  T  is  absent. 

(6)  Where  0  and  T  are  both  present,  P,  R  and 
^  are  also  present. 

(7)  Where  Q  is  absent  while  S  is  present,  R  is 
absent  and  T  present. 

(8)  Where  Q  and  vS*  are  both  absent,  P  and  R 
are  both  absent. 

(9)  W^here  Q  and  T  are  both  absent,  P,  R  and  .S" 
are  also  absent. 

(10)  Where  5  is  absent  while  T  is  present,  P,  Q 
and  R  are  all  absent. 

By  (i).    Everything  is  /  or  Q  or  jR  or  ST', 

By  (ii),  Everything  is  QR  or  qr ; 
therefore,  Everything  is  QR  or  /^^r  or  qrST) 

By  (iii).  Everything  is  ^  or  r  or  j  or  PT  or  // ; 
therefore.  Everything   is  pqr  or  qrSP  or  <2-/?i'  or   PQRT 
OTJ>QRt', 

By  (iv),  Everything  is  ^  or  r  or  6"  or  P/ ; 
therefore,  Everything  is  pqr  or  qrST  or  PQRst  or  PQRST 
01  pQRSt', 

By  (v),  Everything  is  J"  or  P  or  ^  or  i?  or  T^j 
therefore,  Everything  is  pqrs  or  /^r  J*  or  ^^r^'Z'  or  PQRst 
ox  PQRST  ox  pQRSt. 

The  desired  results  follow  from  this  immediately. 

392.     Given  that,— 

A  is  Be  ox  bC; 
B  is  DE  ox  de\ 
Cis  De; 


37(>  COMPLEX   INFERENCES, 

shew  that, — 


[part  IV. 


A  is  BcDE  or  Bcdc  or  bCDe  \ 

BcDisE; 

abd  is  c ; 

cd  is  Be  or  ab\ 

bCD  is  e. 

[Jcvons,  Pure  Logic ^  p.  6C)?[ 

393.  At  a  certain  examination  it  was  observed 
that,— 

(i)  all  candidates  who  were  entered  for  Greek  were 
entered  also  for  Latin ; 

(ii)  all  candidates  who  were  not  entered  for  Greek- 
were  entered  for  English  and  French,  and  if  they 
were  also  entered  for  Latin,  they  were  entered  for 
German ; 

(iii)  all  candidates  who  were  entered  for  Latin 
and  Greek  while  they  were  not  entered  for  English 
were  not  entered  for  French; 

(iv)  all  candidates  who  were  entered  for  Latin  and 
Greek  while  they  were  not  entered  for  French  were 
not  entered  for  German. 

Shew  that, — 

(i)  Every  candidate  was  either  entered  for  Eng- 
lish or  else  for  both  Latin  and  Greek. 

(2)  Every  candidate  was  entered  either  for  Latin 
or  else  for  both  English  and  French. 

(3)  All  candidates  entered  for  French  were  en- 
tered also  for  Eng^lish. 


CHAP.  X.]  COMPLEX   INFERENCES.  377 

(4)  All  candidates  entered  for  German  were  also 
entered  for  both  ILnglish  and  French. 

(5)  If  a  candidate  was  not  entered  for  English, 
he  was  not  entered  for  either  French  or  German,  but 
he  was  entered  for  both  Latin  and  Greek. 

(6)  If  a  candidate  was  not  entered  for  French,  he 
was  entered  for  both  Latin  and  Greek  but  not  for 
German. 

(7)  If  a  candidate  was  entered  for  Latin  and  also 
cither  entered  for  German  or  not  entered  for  Greek,  he 
was  entered  for  English,  French,  and  German. 

(8)  If  a  candidate  was  entered  both  for  Greek 
and  German,  he  was  also  entered  for  English,  Latin, 
and  French. 

(9)  If  a  candidate  was  entered  neither  for  Greek 
nor  German,  he  was  entered  for  English  and  French 
but  not  for  Latin. 

(10)  Every  candidate  was  entered  for  at  least 
two  lancruasres  ;  and  no  candidate  who  was  entered 
for  only  two  languages  was  entered  for  German. 

394.  In  a  certain  year  it  was  obscrv^ed  that  all 
horses  entered  for  the  One  Thousand  were  also 
entered  for  the  Oaks ;  all  horses  entered  for  the  Two 
Thousand  were  also  entered  both  for  the  Derby  and 
the  St  Lcgcr;  all  horses  entered  both  for  the  One 
Thousand  and  the  Derby  were  entered  for  the  Two 
Thousand ;  all  horses  entered  for  the  One  Thousand 
and  the  St  Leger  were  also  entered  for  the  Derby; 
all  horses  entered  either  for  the  Oaks  or  the  St  Leger 
were  entered  either  for  the  Two  Thousand  or  the  One 


378 


COMPLEX   INFERENCES. 


[part  IV. 

Thousand.     Shew  that  in  that  year  all  horses  must 
belong  to  one  or  other  of  the  following  four  classes  :— 

(i)  Horses  entered  for  all  the  following  races, — 
the  Two  Thousand,  the  Derby,  the  Oaks,  the  St 
Leger. 

(2)  Horses  entered  for  none  of  the  following 
races,— the  Two  Thousand,  the  One  Thousand,  the 
Oaks,  the  St  Leger. 

(3)  Horses  entered  for  the  Two  Thousand,  the 
Derby,  and  the  St  Leger,  but  not  for  the  One  Thou- 
sand. 

(4)  Horses  entered  for  the  One  Thousand  and 
the  Oaks,  but  for  none  of  the  three  other  races. 

395.  Shew  the  equivalence  between  the  three 
following  sets  of  propositions  : — 

(i)     A\sD;  (I) 
aB  is  Cde ;  (2) 
Ce  is  ^  or  ^  ;  (3) 
ab  is  C\  (4) 
CDE\sAb\  (5) 
Deis  ABC  ox  Abe  \  (6) 
AbeD  is  e.     (7) 

(ii)     CD  is  A  Be  or  AbB;  (8) 
cD  is  A  ;  (9) 
ci  is  aBCe  or  abCE\  (10) 
AcD  is  BE  ox  be,     (11) 

(iii)     C  is  AD  or  ad\  (12) 
EisbCox  ABcD)  (13) 
e  is  BC  or  AbeD.     (14) 


CHAP.  X.] 


COMPLEX   INFERENCES. 


379 


To  establish  the  desired  result  it  will  suffice  to  shew  that 
(ii)  may  be  inferred  from  (i),  (iii)  from  (ii),  and  (i)  from 
(iii). 
Firsty     (ii)  may  be  inferred  from  (i). 

By  (5),  CDE  is  Ab ; 
therefore,  CDE  is  AbE, 

By  (6),  CDe  is  AB ; 

therefore,  CDe  is  ABe. 

But,  CD\s  CDE  ox  CDe; 

therefore,  CD  is  AbE  or  ABe,  (8) 

By  (2),  aB  is  C] 

by  (4),  ^^  is  C ; 

therefore,  a  \s  C  ] 

therefore,  ^  is  ^  ; 

therefore,  cD  is  A.  (9) 

By  (i),  d\%  a; 
therefore,  d is  aB  or  ab', 
by  (2),  aB  is  aBCe; 
by  (3)  and  (4),  ab  is  ahCE\ 
therefore,  d  is  aBCe  or  abCE.  (10) 

By  (6),  AcD  is  be  ox  E\ 
by  (7),  AcD  \%  B  ox  e) 
therefore,  AcD  is  BE  ox  be,  (11) 

Secondly,     (iii)  may  be  inferred  from  (ii). 

•      By  (8),  CD  is  AD ; 

by  (9),  G/is  ad] 
dierefore,  C  \s  AD  ox  ad.  {12) 

By  (8),  E  is  Ab  ox  c  ox  d ; 

by  (9),  ^  is  ^  or  Cor  d; 

therefore,  E  is  Ab  or  dor  Ac; 

by  (10),  E  is  abC  ox  D ; 

therefore,  E  is  AbD  or  abCd  or  AcD; 


3So  COMPLEX   INFERENCES.  [part  I  v. 

by  (ii),  E  IS  B  or  a  ox  C  or  d\ 
therefore,  E  is  AbCD  or  abCd  or  ABcD ; 

therefore,  E  \%  bC  or  ABcD.  (13) 

By  (8),  c  is  AB  or  c  or  d\ 

by  (9),  ^  is  ^  or  C  or  d) 

therefore,  e  is  AB  or  Ac  or  d\ 

by  (10),  r  is  aBC  or  Z> ; 

therefore,  e  is  ^^Z>  or  AcD  or  aBCd) 

by  (i  i),  ^  is  /^  or  rtJ  or  C  or  d\ 

therefore,  ^  is  ABCD  or  y^/'rZ)  or  aBCd) 

therefore,  ^  is  -^C  or  AbcD.  (14) 

Thirdly^     (i)  may  be  inferred  from  (iii). 

By  (13),  ^^is  Z>; 
by  (14),  cc  is  Z>; 
therefore,  r  is  /^ ; 
therefore,  ^<:  is  D ; 
by  (12),  ^Cis  Z>; 
therefore,  A'\s  D,  (i) 

By  (i3)j  aBise; 
by  (14),  ^^is  C; 
by  (12),  aCis  d ; 
therefore,  aB  is  Cde.  (2) 

By  (14),  Ce  is  /? ; 
therefore,  Ce  is  ^  or  7?.  (3) 

By  (14),  ab  is  E ; 
by  (13),  /;ii  is  C; 
therefore,  ^^  is  C.  (4) 

By  (12),  Ci9is^; 
by  (13),  CE  is  b; 
therefore,  CB>E  is  Ab.  (5) 


CHAP.  X.] 


COMPLEX   INFERENCES. 


381 


By  (14),  E>e  is  BCB>  or  y^/^^; 

by(i2),  ^CZ>is^i?C; 
therefore,  Z>e  is  y^^C  or  Abe.  (6) 

By  (13), /^(Tw^; 
therefore,  ^<^^Z>  is  e.  (7) 

The  desired  conclusion  might  also  be  reached  by  shew- 
ing that  each  set  of  propositions  may  be  summed  up  in  the 
single  proposition, — 

Everything  is  ABCDc  or  ABcDE  or  AbCDE  or  AbcDe  or 

aBCde  or  abCdE, 

396.  Shew  the  equivalence  between  the  two  follow- 
ing sets  of  propositions  : — 

(i)     A  is  BC  or  BE  or  CE  or  D  \ 
B  is  ACDE  or  A Cdc  or  cdE  ; 
C  is  AB  or  ^-C"  or  aD ; 
Z>  is  ABCE  or  ^^^  or  (2(7; 
^  is  ^  C  or  aCD  or  ^r. 

(2)     ^  is  BcdE  or  ^r</^  or  bD ; 
^  is  rt:  or  ce  or  <:/^  ; 
c  is  AbDe  or  rt'^^i?  or  BdE\ 
d  is  <7/^^^  or  ^6-^  or  ^^  or  bE  ; 
^  is  ^^  or  be  or  ^. 

397.  Shew  that  the  following  sets  of  propositions 
are  equivalent : — 

(i)     A  is  BCDE  or  BCd  or  bCD  or  bdE  ; 
C  is  ABde  or  AbDe  or  rt;^^  or  i5Z?^  ; 
E  is  yi^^iT^  or  aBC  or  ^6Z>. 


382 


COMPLEX  INFERENCES. 


[part  IV. 


(2)  AbCDise) 
A  Cd  is  Be ; 
ah  is  c ; 
BCe\s.Ad\ 
CIS  A  bdE  or  ae, 

(3)  ABCdlse] 
ABd  is  C\ 
AbC  IS  D; 
Abed  IS  E\ 

ad  is  BCE  or  r^ ; 

D  \s  AbCe  ox  aee  or  ^C^. 


398.  Shew  the  equivalence    between  the  two 
following  sets  of  propositions  : — 

(i)    ^  is  Bcde  or  bedE\ 
b  is  Acde  or  acdE  ; 
C  is  DE  ; 
eE  is  <7^^; 
D  is  Cii. 

(2)    ^i?is  CD E  or  ede; 
ab  is  r^/i5 ; 
C  is  ABDE', 
D  is  ^i?(r£" ; 
E  is  yi^(:'i?  or  ^^r^. 

399.  Find  which  of  the  following  propositions 
may  be  omitted  without  any  limitation  of  the  informa- 
tion given  : — 

Pq  is  7'S  T ; 
Pr\s  qST] 


CHAP.  X.] 


COMPLEX  INFERENCES. 

Ps  is  QRf ; 

Pt  is  QRs ; 

pT  IS  qr\ 

QisR; 

Qs  is  PRt] 

qS  is  rT'y 

qT  is  PrS  or pr  \ 

RisQ; 

RsisPQf', 

rS  is  qT  'y 

rT  is  PqS  or  pq  \ 

St  is  PQR  QTpqr, 


3^3 


CHAPTER  XL 

PROBLEMS   INVOLVING   SIX   OR   MORE  TERMS. 

400.     Given— (i)    A B  is  CD  or  Ef, 

(ii)     D  is  AbeF, 
(iii)   d'ls  A  or  be, 

(iv)   dE  is  ABC) 
then, — 

(i)  BlsACdEf, 

(2)  rt:  is  bcde, 

(3)  EisABCdf, 

(4)  C  \s  Ab  or  AdEf, 

(5)  bf'isde, 

(6)  ^  is  ^^^  or  ^-^^6', 

(7)  -^  is  ^(T^^/  or  ^Z>^/^  or  <5^r, 

(8)  ^  is  y^  ^^  or  abed  or  yi  Z^Z>i^, 

(9)  F  is  rt-^c^/-^  or  Abe, 

(10)    /is  y^^^^  or  ABCdE  or  ^^(:<:/<?. 

(i)  By  (ii),  .5  is  ^; 

by  (iii),  Bd  is  ^  ; 
therefore,  B  is  -^^ ; 


CHAP.  XL]  COMPLEX  INFERENCES. 

by  (i),  ABd  is  Ef', 
therefore,  B  is  AdEf; 

by  (iv),  dE  is  C; 
therefore,  B  \s  A  CdEf. 


(^) 


(3) 


(4) 


(5) 


(6) 


(7) 


By  (ii),  a\%  d\ 
by  (iii),  ad  is  ^^; 

therefore,  a  is  ^^^; 
by  (iv),  M  is  ^ ; 

therefore,  ^?  is  bcde. 

By  (ii),  ^  is  ^; 

by(iv),^/^is^^C; 
therefore,  E  is  ABCd; 

by  (i),  ^^//  is/; 
therefore,  -£  is  ABCdf. 

By  (i),  C'ls  b  or  AdEf; 
by  (2),  C  is  ^  ; 
therefore,  C  is  Ab  or  ^//j^^ 

By  (ii),/is^; 

by  (3),  bise; 

therefore,  bf  is  de. 

By  (ii),  ^  is  y4  or  ^; 
therefore,  e\^  A  or  ad; 
by  (i),  CIS  b; 
by  (3),  r  is^; 
therefore,  <r  is  Abe  or  ^<^^<?. 


385 


By  (i),  B  is  A  CdEf; 
by  (ii),  Z>  is  AbeF; 
by  (iv),  ^^  is  ^. 

But,  (by  the  law  of  excluded  middle),  ^  is  ^  or  Z)  or  bd; 
therefore,  A  is  B  CdEf  01  bDcFox  bde. 


K.  L. 


25 


386  COMPLEX  INFERENCES.  [part  iv. 

(8)  By  (2),  a  is  hd] 

by  (ii),  D  is  AbF, 

But,  e  is  Ad  or  a  or  D ; 
therefore,  e  is  Ad  or  adcd  or  AbDF\ 
and  by  (i),  ^  is  ^  j 
therefore,  c  is  y^M  or  abed  or  AbDF, 

(9)  By  (2),  d5  is  d!^^^(? ; 
therefore,  F  is  «2/^^^/^  or  A. 

By  (i)  and  (3),  i^is  ^^'; 
therefore,  i^is  abcde  or  ^^^. 

(10)  By(3),  ^is^; 

by  (i),  i?is^G/^; 
by  (2),  a  is  abcde. 

But,/ is  Ab  ox  B  ox  a\ 
therefore,/ is  Abe  or  ABCdE  or  abcde \ 
and  by  (ii),/is  ^; 
therefore, /is  ^^^/e?  or  ABCdE  or  ^/^r^/<?. 

401.  Three  persons  A^  B,  Cy  are  set  to  sort  a 
heap  of  books  in  a  library.  A  is  told  to  collect  all 
the  English  political  works,  and  the  bound  foreign 
novels :  ^  is  to  take  the  bound  political  works,  and 
the  English  novels,  provided  they  are  not  political:  to 
C  are  assigned  the  bound  English  works  and  the 
unbound  political  novels.  What  works  will  be  claimed 
by  two  of  them  }     Will  any  be  claimed  by  all  three  } 

[Venn,  Symbolic  LogiCy  pp.  264,  265.] 

Let  P^  English, 
Q  =  political, 
J^  =  bound, 
S  =  novel ; 


CHAP.  XL]  COMPLEX   INFERENCES. 


387 


A  =  books  assigned  to  A^ 
B  =  books  assigned  to  Bj 
C=  books  assigned  to  C. 
The  premisses  tell  us  that, — ■ 

(i)    FQisA, 

(2)  />BSisA, 

(3)  G7?is^, 

(4)  F^S  is  B, 

(5)  ^^isC, 

(6)  QrS  is  C; 

and  the  problem  is  to  determine  of  what  we  can  affirm 
respectively  AB,  BQ  CA,  ABC. 

By  (i),  what  is  both  F  and  Q  is  A, 
by  (3),  what  is  both  Q  and  F  is  B ; 

therefore,  what  is  both  F,  Q,  and  F  is  AB.     (i). 

Combining  (2)  and  (3)  similarly,  we  have 

p  QR S  is  AB.     (ii). 

From  (i)  and  (4)  we  get  nothing,  since  nothing  can  be 
F  and  Q,  and  at  the  same  time  F  and  not- (2-  Nor  does 
the  combination  of  (2)  and  (4)  yield  anything.  We  find 
then  that  **  English  bound  political  works  and  foreign  bound 
political  novels  are  claimed  both  by  A  and  BJ' 

Similarly,         FQR  is  BC  (iii) 

F/FS  is  BC.  (iv) 

FQF  is  CA.  (v) 

FQrS  is  CA.  (vi) 

Lastly,  (i)  and  (iii)  give  FQF  is  ABC;  and  it  will 
easily  be  seen  that  FQF  is  the  only  combination  of  which 
this  is  true. 

25—2 


3S8 


COMPLEX    INFERENCES. 


[part  IV. 


402.  (i)  Where  A  or  C  or  £  is  present,  B  or  D 
or  F  is  present,  and  vice  versa; 

(2)  Where  B  is  present  and  C  absent,  or  B  absent 
and  C  present,  D  is  present  and  E  absent  or  D  is 
absent  and  E  present,  and  vice  versa  ; 

(3)  Wliere  both  A  and  D  are  present,  F  is  absent ; 

(4)  Where  D  is  present,  E  is  absent,  and  vice 
versa; 

(5)  Where  C  is  present,  D  is  absent. 

Shew  that  where  C  is  present,  B  is  absent,  and 
vice  versa,  [Jcvons,  Pure  Logic,  pp.  66y  ^J^ 

By  (2),   (De  is  Be  or  hC^ 
\dE  is  Be  or  I? C ; 
but  by  (4),  Everything  is  De  or  dB ; 
therefore,  Everything  is  ^^  or  ^C; 
therefore,  I  C  is  /^, 


iC  IS  /', 
^  is  C. 


403.     Given  the  same  premisses  as  in   the   pre- 
ceding example,  shew  that, — 

(i)     Where  C  and  D  are  both  absent,  B  and  E 
are  both  present,  and  vice  vet^sd  ; 

(2)  Where  D  is  present,  A  and  B  are  present, 
while  C,  E,  and  /^  are  absent,  and  vice  versa ; 

(3)  Wliere  E  is  absent,  ^,  B,  and  Z^  are  present, 
while  C  and  i^  are  absent,  and  vice  versa; 

(4)  Where  B  is  absent,  6^,  ZT,  and  /'^  are  present, 
while  D  is  absent ; 

(5)  Where  D  and  /^  are  both  absent,  B  and  E 
are  present  while  6^  is  absent. 

[Jevons,  Pin^e  Logic,  pp.  66,  6y.'\ 


CHAP.  XL]  COMPLEX   INFERENCES.  389 

404.  With  respect  to  certain  classes  of  pheno- 
mena, it  is  observed  that, — 

(i)  Wliere  B  is  absent,  E  is  present,  but  C,  D,  and 
F  are  absent ; 

(ii)  Where  B  is  present  while  D  is  absent,  A,  C, 
and  E  are  present,  but  F  is  absent ; 

(iii)  \{ B  and  D  are  both  present,  E  is  not  present 
F  being  absent,  nor  is  C  present  A  being  absent. 

It  may  hence  be  deduced  that, — 

(i)     If  ^  is  absent,  ^'is  absent. 

(2)  If  ^  is  present,  either  C  or  D  is  present, 

(3)  If  i>,  D,  and  E  are  all  present,  F  is  present. 

(4)  If  6'  is  absent,  B  and  D  are  either  both  pre- 
sent or  both  absent. 

(5)  If  C  and  D  are  both  absent,  B  is  absent. 

(6)  \i  C  is  present,  A  and  B  are  both  present. 

(7)  If  Z^  is  present,  ^  is  present. 

(8)  If  Z>  is  absent,  E  is  present  but  F  is  absent. 

(9)  If  F  is  absent,  D  and  E  cannot  both  be  pre- 
sent or  both  absent. 


405.     Given, — 

(i)  aB  is  c  or  D\ 

(2)  BE  is  DF  or  cdF; 

(3)  C  is  aB  or  BE  or  D  ; 

(4)  bD  is  e  or  F) 

(5)  bf  is  a  or  C  or  DE  \ 

(6)  bcdE  is  Af  or  aF ; 

(7)  ^  is  i?  or  ^'i^i?!/  or  cDf  or  r^/£" ; 


390 


COMPLEX   INFERENCES.  '         [part  IV. 


it  follows  that, — 

(i)  ^  is  ^  ; 
(ii)  CisD; 
(iii)   E  is  F, 

406.  One  season  at  a  certain  hotel  in  Switzerland 
it  happened  that  all  the  visitors  were  either  English 
or  Americans;  all  who  went  on  mountaineering  ex- 
peditions were  either  lawyers  or  English  members  of 
a  University  or  unmarried  American  ladies ;  none  of 
the  lawyers  were  ladies ;  all  the  English  lawyers 
were  members  of  a  University;  all  the  ladies  who 
were  members  of  a  University  were  American  or  un- 
married ;  all  the  Americans  who  were  not  members 
of  a  University  were  married ;  all  the  members  of  a 
University  who  were  not  lawyers  were  mountaineers ; 
the  mountaineers  who  were  members  of  a  University 
were  either  Americans  who  were  not  lawyers  or  else 
ladies. 

Obtain  the  fullest  descriptions  you  can  of  the 
English  mountaineers ;  the  lawyers ;  the  members  of 
a  University;  those  who  were  not  members  of  a 
University ;  the  American  ladies ;  the  American 
mountaineers;  the  unmarried  non-mountaineers;  the 
unmarried  men ;  the  married  men  who  were  not 
lawyers. 

407.     Shew  the  equivalence  between  the  two  fol- 
lowing sets  of  propositions: — 

(i)    ^^is  CD  or  EF) 
Cd  is  Ad  or  Ef\ 


CHAP.  XL]  COMPLEX   INFERENCES.  391 

eF  is  aB  or  cD\ 
ab  is  cd\ 
cd  is  ef\ 
efis  ab, 

(2)   a  is  EC  or  BD  or  bcdef\ 
A  Bis  CDE  or  cDEF\ 
e  is  abcdf  or  F\ 
Abed  is  ef\ 
aBCd\sf\ 
AbCFisE. 

408.     Given, — 

i)    ^^  is  DE  or  Dfor  hi, 

2)  C  is  aB  or  DEFG  or  BFH, 

3)  Bed  is  cK  or  hi, 

4)  Aefxsdy 

5)  i  is  BC  or  Cd  or  Cf  or  H, 

6)  ABCDEFG  IS  H  or  I, 

7)  DEFGHisB, 

8)  ABkisforh, 

9)  ADFIkxsH, 

10)  ADEFH  'is  B  or  Cor  G  or  K\ 
shew  that, — A  is  K, 

This  problem  involves  ten  terms;  and  its  solution  will 
shew  the  power  of  the  methods  that  we  have  been  con- 
sidering. 

It  may  be  solved  in  a  straightforward  manner  by  the 
general  method  formulated  in  section  355  : — 

By  (i),  ^  is  ^  or  C  or  DE  or  Z^or  ///; 

By  (2),  A  is  BFHox  c  or  DEFG', 
therefore,  A  is  Be  or  BFH  or  cDE  or  cD/ox  chi  or  DEFG^ 


392 


COMPLEX   INFERENCES.  [part  iv. 


By  (3),  A  is  b  or  C  or  D  or  Jii  or  K) 
therefore,  A  is  BCFH  or  BcD  or  BDFH  or  cDE  or  cDf 

or  chi  or  DEFG  or  A'; 

By  (4),  A  is  C  or  d  or  F\ 
therefore,  A  is  BCFH  or  BcD F  or  BDFH  or  cDEF 
or  cdhi  or  ^/7//  or  DEFG  or  A'; 

By  (5),  A  is  i?Cor  G/or  Cf  or  H or  /; 

therefore,  ^  is  BCDEFG  or  BCFH  or  BcDFI  or  BDFH 

or  cDEFH  or  rZ>^/7  or  DEFGH  or  DEFG  I  or  A'; 

By  (6),  ^  is  ^  or  r  or  ^  or  ^  or  for  g  or  ^or  /; 
therefore,  ^  is  BCFH  or  BcDFI  or  BDFH  or  cDEFH 
or  r/?^J^/  or  DEFGH  or  DEFG  I  or  ^; 

By  (7),  A'\%  B  or  dor  e  or  for  g  or  //; 

therefore,  ^  is  ^'C/7/  or  BcDFI  or  B  DEFG  I  or  BDFH 

or  cDEFgH  or  cDEFgl  or  cDEFhl  or  DEFGhl  or  A'; 

By  (8),  ^  is  Z^  or/  or  //  or  AT; 

therefore,  ^  is  BcDFJiI  or  bcDEFgll  or  bcDEFgl 

or  cDEFhl  or  DEFGhl  or  A'; 

By  (9),  ^  is  ^  or/ or  //or  /  or  A"; 
therefore,  ^  is  bcDEFgH  or  AT; 

By  (10),  ^  is  -5  or  C  or  //  or  ^  or/or  C7  or  //  or  A'; 

therefore,  -<4  is  K, 

The  problem  may  also  be  solved  as  follows  : — 
By  (9),  ADF  is  AT  or  AT  or  /; 

By  (6),  A  BCDEFG  is  H  or  /; 
therefore,  A  BCDEFG  \s  H  or  K', 

By  (S),  ABE  is /i  or  X; 
therefore,  ABCDEFG  is  K, 
therefore,  No  ABCDEFG  is  k; 
therefore,  No  ABCk  is  DEFG.  (i) 


CHAP.  XL]  COMPLEX   INFERENCES. 


393 


By  {2\AC is  DEFG  or  ^Z^^; 

But  by  (8),  ^Z:^  is  a  or  A"; 

therefore,  ^CX'  is  DEFG] 

therefore,  by  (i),  No  ACk  is  ABCk] 

therefore,  ^o  AC  is  Bk; 

therefore,  ABC  is  K. 

By  (3),  ^r^/  is  K  or  /;/ ; 
But  by  (5),  ///  is  C\ 
therefore,  Bed  is  A". 

By  (9),  ADFk  is  AT  or  /; 
By  (5),^  is /Tor/; 

therefore,  AcDFis  H  or  A"; 
By  {Z\  ABF  is  h  or  K] 
therefore,  A  BcD F  is  K; 

therefore,  A  BcD  is/ or  A"; 

By  (4),  Acf  is  d] 

therefore,  AcD  is  F] 

therefore,  ABcD  is  K] 

But  by  (iii),  y^^r^  is  K] 

therefore,  ABc  is  K; 

And  by  (ii),  ^^C  is  K] 

therefore,  AB  is  K. 

By  (5),  ///  is  ^C  or  Cd  or  (7"; 

therefore,  bDF  is  H  or  /; 

By  (9),  ADF  is  H  or  i  or  A"; 

therefore,  AbDF  is  AT  or  AT". 

By  (7),  bDEFisgorh] 

therefore,  AbDEF  is  g  or  A"; 

therefore,  AbDEFG  is  A'; 

therefore,  DEFG  is  a  or  B  or  A"; 

By  (2),  ^/;Cis  A)AA6^; 

therefore,  AbC  is  A". 


(ii) 


(iii) 


(iv) 


(v) 


(vi) 


394 


COMPLEX   INFERENXES. 

By  (lo),  AbcDEF  IS  G  or  //  or  A'; 

By  (7),  bDEF'i^  g  or  h] 

therefore,  AbcDEF  is  h  or  K\ 

By  (v),  AbDF'xs  H  or  K] 

therefore,  AbcDEF  is  K) 

therefore,  AbcDE  is /or  K-j 

By  (4),  AcD  is  F', 
therefore,  AbcDE  is  K, 

By  (i),  be  is  Z>^  or  Df  or  hi) 

and  by  (5),  be  is  //or  /; 

therefore,  be  is  /^jfi"  or  /y"; 

By  (4),  Ac  is  d^  or  F) 
therefore,  Abe  is  DE\ 
therefore,  Abe  is  AbcDE \ 
and  therefore,  by  (vii),  ^/;^  is  K\ 

But  by  (vi),  ^^Cis  A'; 

therefore,  Ab  is  A"; 

and  by  (iv),  AB  is  A'; 

therefore,  A  is  AT. 


[part  IV. 


(vii) 


CHAPTER  XII. 


INVERSE   PROBLEMS. 


409.     Nature  of  the  Inverse  Problem. 

By  the  Inverse  Problei7i  I  mean  a  certain  problem  so- 
called  by  Professor  Jevons.  Its  nature  will  be  indicated  by 
the  following  extracts,  which  are  from  the  Principles  of 
Science  and  the  Studies  in  Deductive  Logic  respectively. 

*'In  the  Indirect  process  of  Inference  w^e  found  that 
from  certain  propositions  we  could  infallibly  determine  the 
combinations  of  terms  agreeing  with  those  premisses.  The 
inductive  problem  is  just  the  inverse.  Having  given  cer- 
tain combinations  of  terms,  we  need  to  ascertain  the  pro- 
positions with  which  they  are  consistent,  and  from  which 
they  may  have  proceeded.  Now  if  the  reader  contemplates 
the  following  combinations, — 

ABC  abC 

'  aBC  abe, 

he  will  probably  remember  at  once  that  they  belong  to  the 
premisses  A=AB,  B  =  BC.  If  not,  he  will  require  a  few 
trials  before  he  meets  with  the  right  answer,  and  every  trial 


396 


COMPLEX   INFERENCES. 


[part  IV. 


will  consist  in  assuming  certain  laws  and  observing  whether 
the  deduced  results  agree  with  the  data.  To  test  the  facility 
with  which  he  can  solve  this  inductive  problem,  let  him 
casually  strike  out  any  of  the  possible  combinations  involving 
three  terms,  and  say  what  laws  the  remaining  combinations 
obey.  Let  him  say,  for  instance,  what  laws  are  embodied 
in  the  combinations, — 

ABC  aBC 

Abe  abC, 

*'  The  difficulty  becomes  much  greater  when  more  terms 
enter  into  the  combinations.  It  would  be  no  easy  matter 
to  point  out  the  complete  conditions  fulfilled  in  the  com- 
binations,— 

ACe 

aBCc 
aBcdE 
ahCe 
abcE. 

After  some  trouble  the  reader  may  discover  that  the  prin- 
cipal laws  are  C=^e,  and  A  =  Ae\  but  he  would  hardly 
discover  the  remaining  law,  namely  that  BD^BDc'' 
{Principles  of  Science,  isted.,vol.  i.,  p.  144;  2nd  ed.,  p.  125.) 

**The  inverse  problem  is  always  tentative,  and  consists 
in  inventing  laws,  and  trying  whether  their  results  agree 
with  those  before  us."    {Stutiics  in  Deductive  Logic,  p.  252.) 

I  should  myself  rather  prefer  to  state  the  problem  as 
follows  : — 

Given  a  single  proposition  of  the  form, — 

Everything  is  P^or  P^ or  P^\ 

to  find  a  set  of  propositions  involving  as  simple  relations  as 
possible  which  shall  be  equivalent  to  it. 


CHAP.  XII.]         COMPLEX   INFERENCES. 


397 


It  is  strictly  true  that  the  inverse  problem  is  indetermi- 
nate, for  since  we  may  find  a  number  of  sets  of  propositions 
which  are  precisely  equivalent  in  logical  force,  any  inverse 
problem  will  admit  of  a  number  of  solutions.  But  I  do  not 
think  that  it  is  necessary  in  order  to  solve  any  inverse  pro- 
blem to  have  recourse  to  a  series  of  guesses,  nor  that  the 
method  of  solution  need  be  described  as  tentative.  In  the 
following  section,  I  give  what  appears  to  be  an  easy  rule 
for  finding  a  fairly  satisfactory  solution  of  any  inverse 
problem.  Since,  however,  a  number  of  solutions  are  possi- 
ble, some  of  v/hich  are  simpler  than  others,  the  process 
must  be  regarded  as  tentative  so  far  as  we  seek  to  obtain 
the  most  satisfactory  solution. 

We  can  hardly  lay  down  any  absolute  standard  of  sim- 
plicity ;  but  comparing  two  equivalent  sets  of  propositions, 
we  may  generally  speaking  regard  that  one  as  the  simpler 
which  contains  the  smaller  number  of  categorical  propo- 
sitions ^  If  the  number  of  such  propositions  is  equal,  then 
I  should  count  the  number  of  terms  involved  in  their  sub- 
jects and  predicates  taken  together,  and  regard  that  one  as 
the  simpler  which  involves  the  fewer  terms.  If  we  have  to 
compare  disjunctives  with  categoricals,  we  may  regard  a 
proposition  with  two  alternatives  in  the  predicate  as  equiva- 
lent to  two  categorical  propositions,  one  with  three  alterna- 
tives as  equivalent  to  three  categorical  propositions,  and 
so  on*. 


^  By  a  categorical  proposition  here  I  mean  one  which  does  not 
involve  disjunctive  combination,  (although  it  may  involve  conjunctive 
combination),  either  in  the  subject  or  in  the  predicate. 

'•^  When  Professor  Jevons  speaks  of  the  extreme  difficulty  of  the 
inverse  process,  he  apparently  has  in  view  a  resolution  into  a  small 
number  of  categorical  propositions;  and  at  this  I  have  aimed  in  my 
solutions  of  inverse  problems. 


398 


COMPLEX   INFERENCES. 


[part  IV. 


410.  A  General  Solution  of  the  Inverse  Problem, 
— i.e.,  Given  a  proposition  limiting  us  to  a  number  of 
complex  alternatives  to  find  a  set  of  propositions  in- 
volving as  simple  relations  as  possible  which  shall  be 
equivalent  to  it. 

The  data  may  be  written  in  the  form, — 

Everything  is  /*  or  (7  or  5  or  7"  or  &c., 
where  -P,  (2,  ^'c.,  are  complex  terms. 

By  contraposition*  we  may  bring  over  one  or  more  of 
these  complex  terms  from  the  predicate  into  the  subject, 
so  that  we  have, — 

What  is  neither  F  nor  5  nor  &c.  is  ^  or  7"  or  &c. 

The  selection  of  certain  terms  for  transposition  in  this 
way  is  arbitrary,  (and  it  is  here  that  the  indeterminate- 
ness  of  the  problem  becomes  apparent) ;  but  it  will  gene- 
rally be  found  best  to  take  two  or  three  which  have  as  much, 
in  common  as  possible. 

"What  is  neither  Fr\or  S  nor  &c.  is  ^  or  7" or  &:c." 
will  immediately  resolve  itself  into  a  series  of  propositions, 
which  taken  together  give  all  the  information  originally 
given ^  If  any  of  these  are  themselves  very  complex  we 
may  proceed  with  them  in  the  same  way.  We  may  then 
suppose  ourselves  left  with  a  series  of  fairly  simple  propo- 
sitions ;  but  it  will  probably  be  found  that  some  of  these 
merely  repeat  information  given  by  others,  so  that  they 
may  be  omitted. 

We  may  find  to  what  extent  this  is  the  case,  by  adopting 
any  one  of  the  three  following  methods  : — 

First,  by  leaving  out  each  proposition  in  turn,  and  de- 
termining (by  the  ordinary  rules)  what  the  remainder  by 
combination  give  concerning  its  subject.     If  we  find  that 

^  Cf.  section  317.  -  Cf.  chapter  n. 


CHAP.  XII.]         COMPLEX   INFERENCES. 


399 


it  adds  nothing  to  the  information  that  they  give  it  may  be 
omitted. 

Secondly,  by  bringing  each  proposition  to  the  form, — 
Nothing  is  X,  or  X^ or  X^, 

and   then   comparing   it  with  the  combination  of  the  re- 
mainder. 

Thirdly,  by  writing  down  all  possible  combinations  after 
Jevons's  plan,  {Fiire  Logic,  p.  46;  Frinciples  of  Science, 
Chapter  vi ;  Studies,  p.  181),  and  noting  which  are  excluded 
by  each  proposition  in  turn.  If  a  proposition  excludes  no 
combination  that  is  not  also  excluded  by  other  proposi- 
tions it  may  be  omitted. 

We  are  now  left  with  a  series  of  propositions  which 
are  mutually  independent.  By  further  comparison  however 
we  shall  probably  find  that  some  of  them  may  be  still  further 
simplified.  When  such  simplification  has  been  carried  as 
far  as  possible  we  shall  have  our  final  solution. 

This  may  be  verified  by  recombining  the  propositions  that 
we  have  obtained,  by  which  operation  we  ought  to  arrive 
again  at  the  series  of  alternatives  with  which  we  started. 

To  illustrate  the  above  method,  four  examples  follow 
which  are  w^orked  out  in  full  detail. 

I.  For  our  first  example  we  may  take  one  of  those 
chosen  by  Jevons  in  the  extract  quoted  in  section  409. 

Given  the  proposition  that  "  Everything  is 

ABC 
or  Abe 
QxaBC 
or  abC;' 

find  a  set  of  propositions  involving  as  simple  relations  as 
possible  which  shall  be  equivalent  to  it. 


4CX) 


COMPLEX  INFERENCES.  [part  iv. 


By  contraposition,  What  is  neither  ABC  nor  Abe  is 
aBC  or  abC)  therefore,  What  is  a  or  Be  or  bC  is  aBC  or 

abC  y 

i.e.,      (a  is  C, 

./  ^iT  is  not, 
\bC'\s  a. 
"  ^e;  is  not"  is  reducible  to  "i?  is  C";  and  this  proposition 
and  ''a  is  C"  may  be  combined  into  "c  is  Ab:' 
Our  solution  therefore  is, — 

U  is  ^^, 
(<^C  is  a. 

By  combining  these  propositions  it  will  be  found  that  we 
regain  the  original  proposition. 

II.     We  may  next   take   the   more   complex   example 
contained  in  the  extract  from  Jevons  quoted  in  section  409. 

The  given  alternatives  are, — 

ACe, 
aBCey 
aBcdE^ 
abCe, 
abcE. 
Therefore,  What  is  not  aBcdE  or  abcE  is  ACc  or  aBCe  or 
abCe\ 

therefore,      [A  is  Ce-, 

C  is  Ae  or  aBe  or  abe ; 
^  is  ^C  or  aBC  or  abC; 
BD  is  ACe  or  aCe\ 
these  propositions  are  immediately  reducible  to, — 

^A  is  Ce\ 
Cis  ^; 
e  is  C\ 
^BD  is  Ce ; 


-i 


CHAP.  XII.]         COMPLEX  INFERENCES.  401 

and  they  may  be  further  resolved  into, — 

E  is  ac\ 
c  is  aE\ 
BD  is  Ce, 

This  solution  again  may  be  verified  by  re-combination. 

IIL     The  following  problem  is  from  Jevons,  Principles 
of  Science,  2nd  ed.,  p.  127,  (Problem  v.). 

The  given  alternatives  are, — 

A  BCD, 

ABCd, 

ABcd, 

AbCD, 

AbcD, 

aBCD, 

aBcD, 

aBcd, 

abCd. 

Then,  by   contraposition,  what  is  neither  of  the  four 
following, — 

ABCD, 
ABCd, 
aBcD, 
aBcd, 

must  be  one  of  the  remainder. 

But  "  What  is  neither  ABCD,  ABCd,  aBcD  nor  aBcd;' 
is  equivalent  to  "What  is  neither  ABC  nor  aBc,''  and  that 
is  equivalent  to  *' What  is  b  or  Ac  or  ad' 

Therefore,  What  is  b  ox  Ac  ox  aC  is  ABcd  or  AbCD  or 
AbcD  or  aBCD  or  abCd. 


K.  L. 


26 


402 


becomes 
that  is, 


COMPLEX   INFERENCES.  [part  iv. 

From  this  we  have  our  first  resolution  of  the  given 
information  in  the  three  propositions: — 
(i)     b\s  AD  ox  aCd \ 

(2)  Ac  \?>  Bd  ox  bn -y 

(3)  aC'x^BDoxbd, 

Each  of  these  may  again  be  broken  up  into  two  pro- 
positions : — 

b  is  AD  or  aCd, 

If  b  is  not  AD,  it  is  aCd\ 
(i)     ab  is  6V, 
(ii)     bd  is  aC. 
(2)  may  similarly  be  broken  up  into, — 

(iii)     ABc  is  d, 
(iv)    Acd  is  B ; 
and  (3)  into,—  (v)     aBC  is  Z>, 

(vi)     aCD  is  B. 
But  (iv)  is  inferrible  from  (ii),  and  (vi)  is  inferrible  from 
(i);  (iv)  and  (vi)  may  therefore  be  omitted. 
We  have  then  for  our  final  solution, — 

(i)     ab\%  Cd, 

(2)  bd  is  a Cf 

(3)  ABc  is  d, 

(4)  aBC  is  D. 

This  is  practically  equivalent  to  the  solution  given  in 
Jevons,  Studies^  p.  256. 

We  may  now  verify  it  as  follows: — 

By  (i).         Everything  is  ^  or  ^  or  Cd\ 

By  (2),         Everything  is  aC  ox  B  ox  D\ 
therefore,        Everything  is  AD  or  aCd  or  B ; 


CHAP.  XII.]         COMPLEX  INFERENCES.  403 

By  (3),       Everything  is  ^  or  ^  or  C  ox  d; 
therefore,  Everything  is  AbD  ox  ACD  or  aB  ox  aCd oxBC 
or  Bd; 

By  (4),      Everything  is  A  ox  b  ox  c  ox  D ; 
therefore.  Everything  is  ABC  or  ABd  or  AbD  or  ACD  or 
a  Be  or  aBD  or  abCd  or  BCD  or  Bed 

But,  AbD  is  AbCD  or  AbcD. 

Expanding  all  the  terms  similarly,  we  have, — 

Everything  is  ABCD, 
or  ABCd, 
or  ABcd, 
or  AbCD, 
or  AbcD, 
or  aBCD, 
or  aBcD, 
or  aBcdf 
or  abCd. 

These  are  precisely  the  alternatives  that  were  originally 
given  us. 

IV.  The  following  example  is  also  from  Jevons,  Frhi- 
ciples  of  Science,  2nd  Edition,  p.  127,  (Problem  viii).  In  his 
Studies,  p.  256,  he  speaks  of  the  solution  as  ufiknown  ;  and 
I  am,  therefore,  the  more  interested  in  shewing  that  a  fairly 
simple  solution,  involving  no  more  than  five  categorical 
propositions,  may  be  obtained  by  the  application  of  the 
general  rule  formulated  in  this  section. 

The  given  alternatives  are, — 

ABCDE, 
ABCDe, 
ABCde, 
ABcde, 

26 — 2 


404 


COMPLEX   INFERENCES. 


[part  IV. 


AbCDE, 

AbcdE, 

Abcde, 

aBCDe, 

aBCde, 

aBcDcy 

abCDCy 

abCdE, 

abcDe, 

abcdE. 

Therefore,  What  is  neither  ABCDE  nor  ABCDe  nor 
ABCde  nor  Abcde  nor  aBCDe  nor  aBCdc  is  ABcde  or 
AbCDE  or  AbcdE  or  aBcDe  or  abCDe  or  abCdE  or  ^/^rZ^<? 
or  abcdE, 

Now,  "What  is  neither  ABCDE  nor  ABCDe  nor 
y^^C^^  nor  ^/if^^  nor  ^i?CZ>^  nor  aBCde''  is  equivalent  to 
*'  What  is  either  of  the  following, — 

dE, 

bC, 

bD, 

bE, 

Be, 

cD, 

cE, 

aEf 

ab, 


ac. 


j» 


Moreover, 


bE  is  either  bD  or  dE ; 
cE  is  either  cD  or  dE ; 
t7^  is  either  aE  or  at^^; 
a^  is  either  aE  or  ^r^ ; 
bD  is  either  ^C  or  cD. 


CHAP.  XII.]         COMPLEX   INFERENCES. 


405 


Therefore,  our  proposition  becomes  "  What  is 

dEy 
or  bC, 
or  Be, 
or  cD, 
or  aEy 
or  «<^^, 


or  ^^^, 


is  either 


ABcde, 
or  AbCDE, 
or  AbcdE, 
or  aBcDe, 
or  abCDe, 
or  abCdE, 
or  abcDe, 
or  abcdE^^ ; 

and    this    is    resolvable    into    the    following    set   of   pro- 
positions,— 

(i)  //^  is  ^/^  or  /^^; 

(2)  /^C  is  ^Z>^  or  aDe  or  «^^  j 

(3)  /?r  is  Ade  or  ^Z^^ ; 

(4)  cD  is  dr^ ; 

(5)  aE  is  ^^; 

(6)  abe  is  Z>  j 

(7)  ace  is  />. 

Of  these,  (2)  may  be  broken  up  into, — 

(8)  AbC'isDE', 

(9)  bCDE'isA] 

(10)     bCde  is  non-existent. 

But  (9)  may  be  inferred  from  (5),  and  (10)  may  be  in- 
ferred from  (6)  and  (8);  (8)  may  therefore  be  substituted 
for  (2). 


4o6  COMPLEX   INFERENCES.  [part  iv. 

(3)  may  be   inferred   from  (i),  (4),  and  (7),  and  may 
therefore  be  omitted. 

(i)  may  be  broken  up  into, — 

(11)  AdE  is  bc) 

(12)  BdE  is  non-existent. 

But  (12)  may  be  inferred  from  (5)  and  (ii);  (11)  may 
therefore  be  substituted  for  (i). 

Again,  (6)  and  (7)  may  be  combined  into, — 

(13)  ade  is  BC^ 

which  may  therefore  be  substituted  for  them. 

Our  set  of  propositions  may  therefore  be  reduced  to, — 

(i)     AdE  is  bc\ 
(ii)    AbCxsVE) 
(iii)   cD  is  ae; 
(iv)    aE  is  bd; 
(v)     ade  is  BC. 
From  (iv)  it  follows  that  acD  is  e;  (iii)  may  therefore  be 
reduced  to  cD  is  a. 

From  (ii)  it  follows  that  AbdE  is  c;  (i)  may  therefore  be 
reduced  to  AdE  is  b. 

We  are,  therefore,  left  with  the  following  as  our  final 
solution : — 

(i)    AdE  is  b; 

(ii)  Ab  C  is  Z>E; 

(iii)  cZ>  is  a; 

(iv)  aE  is  bd; 

(v)  ade  is  BC. 

This  solution  may  be  verified  as  follows : — 

By  (i),  Everything  is  ^  or  ^  or  2?  or  ^; 


CHAP.  XII.]         COMPLEX   INFERENCES.  407 

By  (ii),  Everything  is  ^  or  ^  or  r  or  B>E; 

therefore,  Everything  is  a  or  BE)  or  Be  or  be  or  eD 

or  ee  or  EE; 

By  (iii).  Everything  is  «  or  C  or  d; 

therefore.  Everything  is  a  or  BCE  or  BCe  or  Bde  or  bed 

or  CEE  or  ede; 

By  (iv),  Everything  is  A  or  bd  or  e; 

therefore.  Everything  is  A  BCD  or  ACDE  or  ahd  or  ae 
or  BCe  or  Bde  or  bed  or  cde\ 

By  (v).  Everything  is  A  or  j5C  ox  D  or  E) 

therefore.  Everything  is  A  BCD 

or  ^^rt^(? 
or  Abed 
or  ^CZ)^ 
or  Aede 
or  ^/^^^ 
or  ^Z>^ 
or  BCe 
or  ^r^iS". 

But  A  BCD  is  ABCDE  or  ABCDe,  and  so  with  the 
others. 

Expanding  the  terms  in  this  way,  we  have, — 

Everything  is  ABCDE 
or  ABCDe 
or  A  BCde 
or  ABcde 
or  AbCDE 
or  AbedE 
or  Abede 
or  aBCDc 


4o8 


COMPLEX   INFERENCES. 


[part  IV. 


or  aBCde 
or  aBcDe 
or  abCDe 
or  abCdE 
or  abcDe 
or  abcdE, 

These  are  again  the  alternatives  with  which  we  com- 
menced. 

411.  Another  Method  of  Solution  of  the  Inverse 
Problem. 

Another  method  of  solving  the  Inverse  Problem,  sug- 
gested to  me  (in  a  slighdy  different  form)  by  Mr  Venn,  is  to 
write  down  the  original  complex  proposition  in  the  negative 
form,  /*.  ^.,  to  obvert  it,  before  resolving  it.  It  has  already 
been  shewn  that  a  negative  proposition  with  a  disjunctive 
predicate,  may  be  immediately  broken  up  into  a  set  of 
simpler  propositions. 

In  some  cases,  especially  where  the  number  of  destroyed 
combinations  as  compared  with  those  that  are  saved  is 
small,  this  plan  is  of  easier  application  than  that  given  in 
the  preceding  section. 

To  illustrate  this  method  we  may  take  two  or  three  of 
the  examples  already  discussed. 

I.     Everything  is  ABC  or  Abe  or  aBC  or  abC] 

therefore,  by  obversion.  Nothing  is  AbC  or  Be  or  ae-y 

and  this  proposition  is  at  once  resolvable  into, — 

'Ab  is  <r, 
.^is  Ab\ 


{: 


*  The  student  will  immediately  recognize  that  this  is  equivalent  to 
our  former  solution.     Equationally  it  would  be  written  Ab  —  c, 


CHAP.  XII.]  COMPLEX   INFERENCES. 


409 


II.  Everything  is  ACe  or  aBCe  or  aBcdE  or  abCe  or 
abeE  \ 

therefore,  by  obversion.  Nothing  is  AE  or  CE  or  BDE  or 
Ae  or  BcD  or  ee. 

This  proposition  m.ay  be  successively  resolved  as  fol- 
lows : — 

|No  J^is  A  or  C  or  BD; 

(No  r  is  ^  or  BD  or  e. 

' E  is  ae  \ 
E  is  b  or  d ; 
e  is  aE; 
^  is  ^  or  d, 

lE  is  ae'y 
IbD'is  Ce; 
V  is  aE. 

This  is  the  same  solution  that  we  reached  before. 

III.  Everything  is  ABCD  or  ABCd  or  ABed  or  AbCD 
or  AbeD  or  a  BCD  or  aBeZ>  or  a  Bed  or  abCd; 

therefore,  by  obversion,  Nothing  is  Abd  or  bed  or  ABeD  or 
abe  or  abD  or  aBCd ', 

and    this    proposition    may   be    successively   resolved    as 
follows  : — 

No  bd  is  A  0?'  e ) 

No  ABe  is  £> ; 

No  ab  is  e  or  7J ; 

No  aBC  is  d. 


i 


^bd  is  aC'y 
ABe  is  d ; 
ab  is  Cd'y 
aBC  is  D, 


4IO  COMPLEX   INFERENCES.  [part  i v. 

This  again  repeats  our  original  solution.  It  is  curious 
that  in  each  of  the  above  cases  we  should  by  independent 
methods  have  attained  the  same  result. 

412.  It  is  observed  that  the  phenomena  A,  B,  C 
occur  only  in  the  combinations  A  Be,  abC,  and  abc. 
What  propositions  will  express  the  laws  of  relation 
between  these  phenomena  t    [Jevons,  Studies,  p.  219.] 

Everything  is  ABc  or  abC  or  abe.  Noticing  that  ''abC 
or  abc''  is  equivalent  to  ab,  we  have  by  contraposition, 
What  is  not  ab  is  ABc ;  that  is,  What  is  A  ox  B  is  ABc ; 

that  is,     CA  is  Be, 
\b  is  Ac. 

413.  Find  propositions  that  leave  only  the  follow- 
ing combinations, — A  BCD,  ABcD,  AbCd,  aBCd,  abed. 

[Jevons,  Studies,  p.  254.] 

Jevons  gives  this  as  the  most  difficult  of  his  series  of  in- 
verse problems  involving  four  terms.  It  may  be  solved  as 
follows: — 

Everything  is  A  BCD  or  ABcD  or  AbCd  or  aBCd  or 
abed. 

Noticing  that  ABCD  or  ABcD  is  equivalent  to  ABD, 
we  have,  What  is  neither  AbCd  nor  aBCd  is  ABD  or  abed. 

Therefore,  What  is  AB  or  ab  ox  c  ox  D  is  ABD  or  abed, 
and  this  is  resolvable  into  the  four  propositions, — 

{AB  is  D,     (i) 
ab  is  cd,     (2) 
e  is  ABD  or  abd,     (3) 
^D  is  AB.     (4) 
But  by  (4)  D  is  AB,  and  by  {2)  ab  is  d;  therefore  (3) 
may  be  reduced  to  ^  is  Z>  or  ab, 

i.e. J  cd  is  ab* 


CHAP.  XII.]         COMPLEX  INFERENCES.  411 

Our  set  of  propositions  may  therefore  be  reduced  to, — 

'ab  is  D, 
ab  is  cd, 
cd  is  ab, 
\D  is  AB  \ 

414.  It  is  observed  that  the  phenomena  A,  B,  C, 
D,  E,  F  are  present  or  absent  only  in  the  combi- 
nations,—^i^^TZ^is  ABCDcf,  ABCdEf,  ABeDF, 
ABeDef,  aBeDF,  aBeDef,  bedEf.  What  propositions 
will  express  the  laws  of  relation  between  these  phe- 
nomena }  [Jevons,  Studies,  p.  257.] 

Jevons  gives  five  solutions  more  or  less  differing  from 
one  another,  and  all  expressed  equationally.  The  following 
is  still  another  solution  expressed  in  the  ordinary  proposi- 
tional  forms : — 

{BE/ is  Cd, 
b  is  d, 
C  is  AB, 
d  is  Ef. 

415.  Resolve  the  proposition  "  Everything  Is  one 
or  other  of  the  following, — 

ABCDeF, 

ABeDEf, 

AbCDEF, 

AbCDeF, 

AbeDeF, 

^  Written  equationally,    this   solution  would  appear  still  simpler; 

namely, — 

AB=^D, 

ab  =  at. 


412 


COMPLEX   INFERENCES. 


[part  IV. 


aBCDEf, 
aBcDEf, 
abCDeF. 
abCdeFf 
abcDefy 
abcdcfl^ 
into  a  series  of  simple  propositions. 

[Jevons,  Principles  of  Scieiice,  2nd  cd.,  p.  127, 

(Problem  X.).] 
The  following  is  a  solution  : — 

(i)  ABE  is  cDf\ 

(2)  AcDFisbe-, 

(3)  aF'isbCe; 

(4)  bf  is  ace ; 

(5)  disae; 

(6)  ^is  abc. 

This  is  rather  less  complex  than  the  solution  by  Dr  John 
Hopkinson  given  in  Jevons,  Studies ,  p.  256,  namely  : — 

(i)  d  is  ab ; 

(2)  b  is  A  For  ae) 

(3)  A/isBcDE; 

(4)  E  IS  Bf  or  AbCDF', 

(5)  Be  is  A  CDF', 

(6)  abc  is  ef-, 

(7)  abe/is  c. 

It  will  be  a  useful  exercise  for  the  student  to  shew  that 
these  two  sets  of  propositions  are  really  equivalent. 

416.  How  many  and  what  non-disjunctive  pro- 
positions are  equivalent  to  the  statement  that  "  What 
is  either  ^^  or  bC  is  Cd  or  cD,  and  vice  versa  "? 

[Jevons,  Studies,  p.  246.] 


>^. 


*^ 


CHAP.  XII.]  COMPLEX   INFERENCES. 

We  have  given, — 

^Ab  is  Cd  or  cZ>, 


413 


^C  is  Cd  or  rZ>, 
Cd  is  Ab  or  bC, 
cD  is  Ab  or  bC 


(I) 

(3) 
(4) 


{; 


(i)  may  be  resolved  into, — 

{Abc  is  D,        (5) 
XAbD  is  c,        (6) 

(2)  becomes  bC  is  d.  (7) 

(3)  may  be  resolved  into, — 

i a  Cd  is  b,         (8) 
t^C  is  n.         (9) 

(4)  may  be  resolved  into, — 

\ac  is  d,  (10) 

Be  is  d,         (11) 

But  (6)  may  be  inferred  from  (7);  and  (8)  from  (9). 
We  therefore  have  for  our  solution  : — 

I  Abc  is  D, 
bC  is  d, 
BCisD, 
ac  is  d, 
^Bc  is  d. 

417.  The  following  is  a  further  series  of  inverse 
problems,  which  should  be  solved  by  the  methods 
indicated  in  sections  410  and  411. 

In  each  case  we  have  given  a  complex  proposition 
which  it  is  desired  to  resolve  into  a  series  of  relatively 
simple  propositions. 

(i)  Everything  is  A  BCD  or  aBCD  or  aBCd  or 
abCd  or  abcD  or  abed. 


414 


COMPLEX   INFERENCES. 


[part  IV. 


(2)  Everything  is  A  BCD  or  AbCd  or  aBcD  or 
abed, 

(3)  Everything  is  AhCD  or  AbCd  or  .^^^^  or 
^^^^  or  abCD  or  ^^6^^  or  abed, 

(4)  Everything  is  AbeDE  or  aBCd  or  aBCE  or 
^^^^  or  «^i/^  or  ^/^CV  or  ^^^^  or  «<^Z^^  or  abde  or  BedE 
or  bCDe, 

(5)  Everything  is  ABCE  or  y^2?^^/  or  ABeE  or 
ABde  or  ^^^^  or  abCE  or  ^^^^  or  abdE  or  ^^^<?  or 

(6)  Everything  is  ABCDE  or  ABCdE  or  ABcDE 
or  ABeDe  or  ABcde  or  AbCdE  or  yi/;r^<:'  or  aBCDE 
or  aBCde  or  abCDE  or  abcDc. 

(7)  Everything  is  ABDe  or  ABDF  or  -^^Z^^?  or 
yJr^/  or  ^^Z^^  or  aBDF  or  ^<^CZ^  or  ^^O/  or  rt'^rZ?  or 
fi^^^  or  aCDE  or  ^67?^  or  aCdE  or  rt'^V/^  or  rt-rZ?^  or 
aDEF  or  aDEf  or  aDeF  or  ^Z^^r/"  or  BcDF  or  ^r^Z'  or 
beef, 

(8)  Everything  is  AbdE  or  -^/^^  or  ^^Z'  or 
A edcf  or  aBDF  or  ^^{TZ'  or  aCdE  or  ^^t'  or  /^{7Z>^  or 
bCdfoxbDEF. 

(9)  Everything  is  ABCEf  or  -^^^^^  or  aBCdf  or 
aBedE  or  aBedeF  or  <2^^  or  ^^^'Z. 

(10)  Everything  is  ABeE  For  ABDEF  or  AbCdcf 
or  Abedef  or  AbedF  or  /^/^Z>  or  yJ^^Z"  or  AbdeF  or 
abCef  or  rtZf  or  aBCd  or  aBCDe  or  aBCDEf  or 
abdef. 


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